Also not dealing explicitly with the Victorian era but I will offer this about Don Quixote, widely considered to be the first modern novel. Cervantes first published a very small portion of what was to become Don Quixote and it was wildly popular. So popular, in fact, that people began writing their own apocryphal Quixote stories. Cervantes blasts the apocryphal Quixote tales and authors throughout part 2, and especially in the introduction. Edit So we can say that fan fiction has existed since fiction has. As has been rightfully pointed out, fiction is older than novelization.

One thing that comes to mind from the 18th century: people were upset that Samuel Richardson killed off the main character in Clarissa, so wrote alternative endings in which she 'lived happily ever after' (~1740s).

One prominent example would be the early Sherlock Holmes fandom, although both the original sources and its' early fanfiction span the divide between the Victorian and Edwardian eras. The original run of A.C. Doyle's Sherlock stories end in 1893, and parodies appeared the same year (one of which, by J.M. Barrie, impressed ACD so much that he included it in his autobiography), but the first examples of non-parody fanfic (often stories 'filling in' cases which were offhandedly mentioned in canon but never actually told by ACD) appeared in the 1920s. The Sherlockians were one of the first 'modern,' organized fandoms as we understand them--they had fanzines, anthologies, and conventions in much the same way that 20th century sci-fi fandom operated.

www.dailydot.com/opinion/fic-excerpt-anne-jamison-sherlock-holmes/

TL;DR: I am not familiar with specific works of fanfiction in the Victorian era, but I am familiar with works comparable to fan fiction in more modern times and in pre-Victorian times, which strikes me as odd and so after some discussion within the post itself, I identify as problematic the necessity for a definition of the term 'fanfiction' and differentiate it from Literary Allusions and general fiction especially science fiction and horror (two genres I'm especially acquainted with historical examples of) and even earlier works that we would now consider to be part of a mythology. I still present my answer anyways though as it may inform later discussion or at least partially help you.

So, obligatory I Am Not a Historian, but I technically do have college credit for 2 courses on the history of English Lit in the United States, so I'm going to give this a go. I've also got familiarity with a roughly equivalent time period of British Lit, and admittedly spotty but still decent knowledge of major works going back to Greek Plays (from another 3 or so classes on historical literature in addition to some on American Lit, a parallel pair of classes on British Lit, and at least some knowledge of some more modern literature including post-Victorian science fiction....) I wish I could give you good sources, but this was all years ago and I've done my best to stick to statements and descriptions that should be findable in a variety of sources besides my memory and wikipedia (since neither is a great source) and nobody else has answered, if you ask for specific sources, I'll do my best to find them!

So most of the examples I can think of predate the Victorian era or are arguably on the borderline. So first, the most famous example I can think of isn't exactly fan-fiction per se, but based upon a quick check on wikipedia, the story about the origin of Frankenstein may interest you. Mary Shelley is said to have come up with the idea for the novel Frankenstein while part of a very literary group on vacation including the famous poet Lord Byron. The way I've heard it told, there was a sort of 'spooky story' contest and a lot of the elements and ideas in Frankenstein originated during that contest. This example does show that, just a decade or two before the formal start of the Victorian era in the early 1830's, that there were "notable social circles" engaged in writing fiction, in this example though you have what is considered to be the birth of science fiction and not fan fiction proper, but as it's in approximately the time period you want and involves a literary circle and writing fiction based at least loosely upon other things (science, history, and the romantic movement versus specific other works of fiction) you might be interested.

Another revolutionary work of literature around the time of the Victorian era would be Walt Whitman's Leaves of Grass. Once again, maybe not fan fiction, but an example of poetry that was explicitly rejecting common literary concepts behind published works. Inspired by the new movements evolving out of Romanticism, Leaves of Grass was described as "obscene" by some and I think might serve as a good example of how at least one society in the Victorian era (the United States) did not kindly treat a radical work including overt sexuality and a lack of respect for traditional form, an accurate description of large amounts of fan fiction and often times what first comes to mind when referring to fan fiction.

The narrative about literature at the time of the Victorian era is complicated, but one important thing to note is that the romantic movement was often considered to be peaking around the time of the start of the Victorian era in the 1830s, but was spinning off other movements and receding in importance by the 1850's while American literature began to be more... self-aware and separate from British literature. At the same time, with the winding down of Romanticism you had the rise of the novel (at least in English literature) and more famous poetry and plays of the era (at least that I'm familiar with) come mostly from the late Victorian era. Most works that I can think of that would be more directly comparable to fan-fiction pre-date or post-date the Victorian era: what isn't too difficult to compare to simply popular retellings of well-known tales often seems to have been written before the Victorian era (I would guess this might be blamed on the printing press and eventually popular literature becoming, well, a lot more popular, the eventual end of literature comparable to either a series of chain letters between a bunch of well-read upper-class figures or as only the most palatable of popular and widespreading writings and the divide between them) or basically in response to the new corpus of writings that flourished during the Victoria era up until some arbitrary point in the early 1900's (generally between the start of WW I and the end of WW II).

I guess an important distinction to make now is the difference between fanfiction and simply literary allusions. The line distinguishing fanfiction from other writings is, to the best of my knowledge, a post-Victorian creation. Drawing on "The Bible Unearthed" and "Who Wrote the Bible" and what I might call general knowledge about the origins of "The Epic of Gilgamesh", for some examples, you have two works which appear to have been assembled by multiple authors over hundreds of years oftentimes with explicit reference or lack there of to earlier writings which would later be compiled into one work. For "The Epic of Gilgamesh" I've had a professor jokingly compare some parts to being fanfiction of earlier parts, which makes sense as it's generally accepted that earlier stories were combined into one tale and then expanded upon (and that combined version was likely expanded upon at least once or twice) to create what is in modern times considered to be "The Epic of Gilgamesh". That is, one work manages to combined pretty directly a concept associated primarily with fan fiction: a story told by a second author, potentially from different cultures, era, and places, expanding upon the work of an earlier first author to create a world of fiction with that of a seemingly original story (whatever part of the Epic you'd draw the line at and say everything before it is the original material and everything after is fan fiction). While comparing the Bible to fan fiction is dangerous ground to tread, you can see the key element of later authors taking earlier works and spinning off 'new stories' which are understood be set in the same 'universe' as the earlier works and might even be considered as having retcons and changes which goes directly against stuff from the earlier works.

So, while I think Frankenstein and Leaves of Grass include elements that might be comparable to the creation of modern fanfiction, it's a tricky question because, at least for the works I'm familiar with, the line between "literary allusion" and "cross-over fanfiction" can be.... a little bit thin, I guess? To use that most useful tool of comparison: works around the time of the Victorian era including Frankenstein and Dracula reference history and mythology as well as more 'recent' ideas of their time to create stories we consider to be 'science fiction' and 'horror' which might be seen as separate from 'fan fiction', but the fantastical elements of Frankenstein include references to 'galvanism' and the potential to raise a corpse through electric shock, something that we'd obviously consider fictional now, but something which had a very distinct ring of science when it was written. Why is this important? Well, going back to my earlier mentions, if the authors of later portions of 'The Epic of Gilgamesh' or 'The Bible' considered the stories that created the universe they were writing in to be, at least largely, non-fiction, then is my (or my professor's or those of /r/atheism) comparison of those later works to fan fiction for the earlier works truly fair when the original authors were potentially writing less in a fictional universe and more in the real world they believed they lived in.... a possibly not purely true but not entirely fictional tale set in the 'real' world with historic references sounds to me more like Frankenstein or Dracula than "Harry Potter and the Methods of Rationality".

This leads me to conclude that the very rough working definition of 'fanfiction' I was trying to use is problematic. It seems to basically hinge upon what authors at the time believed, as some examples I would consider as possible early examples of 'fanfiction' might be more comparable to 'science fiction' and so the continuing boom of literature throughout the Victorian Era may simply have created a lot of works whose antecedents are so strongly overshadowed that they are not well-known, but since such antecedents are a necessary (if not sufficient) part of a definition for fanfiction. That is, I can note various works in the Victorian era as having fan fiction, but, perhaps due to my own biases, I don't recognize any as fan fiction, but rather, at most, full of literary allusions).

William Makepeace Thackeray wrote a sequel to Ivanhoe that could definitely be seen as "fanfiction" Rebecca and Rowena was written in 1850, only 30 years after Ivanhoe and could essentially be described as a continuation of the story, but with the ending that Thackeray thought would be better than Scott's. At the end of Ivanhoe the chivalrous knight Ivanhoe marries the "perfect lady" Rowena- causing some fans to be displeased as Rebecca was thought of by them to be far more intelligent, interesting, and an all around better match. Thackeray, in Rebecca and Rowena addresses fan outrage and creates a new ending for Ivanhoe, where he realizes that Rowena is boring and frigid and, through some ridiculous and unlikely plot twists, reconnects with Rebecca.

Jean-Baptiste Cousin de Grainville's Le Dernier Homme (1805) about the last fertile man on Earth, Omegarus, and the last fertile woman, was fanfic'd by Le Dernier Homme, poème imité de Grainville (The Last Man, a Poem Inspired by Grainville) (1832) by Auguste Creuzé de Lesser; L’Unitéide ou la Femme messie (The Unitéide, or the Female Messiah) (1858) by Etienne-Paulin Gagne; and Omégar ou le Dernier Homme (Omegare, or the Last Man) (1859) by Gagne's wife, Élise Gagne.

In a similar vein, in 1775, Johann Wolfgang von Goethe, a famous German author, wrote, among many famous works, a story called "Das Leiden des Jungen Werthers", or "The Suffering of Young Werther".

Spoiling the story, it follows the love-struck protagonist Werther whose love sickness and inner toil lead him to suicide. in the few years following the book's publication, cult groups of young scholars in Germany became so captivated with the story of Werther that many committed suicide in the same vein as Werther. They would also congregate to speculate and obsess over the details and plot points surrounding the story and Werther.

Here's just one of many excerpts detailing the phenomenon: https://books.google.com/books?id=H6mVKvjMMY4C&pg=PA123&lpg=PA123&dq=das+leidendas+leiden+des+jungen+werthers+suicide+scholars&source=bl&ots=ZPfIxbWIdn&sig=u_BBsiuCiSY-PQgAnMX3gJLu3Lk&hl=en&sa=X&ved=0CDEQ6AEwA2oVChMIsLq3o8nPyAIVgTs-Ch0xFAnM#v=onepage&q=das%20leidendas%20leiden%20des%20jungen%20werthers%20suicide%20scholars&f=false

I see that people are posting about things other than V.E. literature, so I will bandwagon. If you look at Arthurian texts (Malory, Chretiens, etc), and start piecing together stories authored by different people but spun in similar ways, you'll notice that it's all just fanfiction. Everywhere. People liked the story of, say, Sir Gawain, so they wrote their own version. I can provide better examples if desired, but I just wanted to toss this in in case it helped.