I don't know if he had the most dissents, but John Marshall Harlan is often called the "great dissenter", mostly referring to him being the sole (and very passionately) dissent in the civil rights cases following the American Civil War. These were the cases that decided the "separate but equal" idea and that federal rights are not necessarily applied to state laws.

I would even read his dissent in Plessy v. Ferguson (it's not that long) in full if you are interested.

http://chnm.gmu.edu/courses/nclc375/harlan.html

At least one other justice, Oliver Wendell Holmes, was also called The Great Dissenter. See e.g. this book about Holmes with that title.

I'm not sure you can categorize dissents purely by looking at the number of them; more important is the weight of the dissent. The most significant dissents could be those that are singularly memorable or those that eventually changed the course of the law.

1. Justice Holmes (along with Justice Brandeis) issued significant dissents in the free speech arena. Around WWI, the Sedition Law was passed, which effectively made it a crime to criticize the government. Justice Holmes was one of the first SCOTUS folks to argue the First Amendment prohibited such things. This is a really long and interesting topic, meriting separate discussion. The dissent is available here. Free Speech in its Forgotten Years catalogs the change in the law.

2. The modern 'great dissenter' is probably Antonin Scalia, who is famous for pithy dissents. There is a school of thought that he has slowly taken over the intellectual reins of the conservative movement (see e.g. this article). Justice Scalia's dissents include, recently, a tirade about immigrants - see e.g.this article, and about gay marriage, see Lawrence v. Texas, or in general see this summary from the New Yorker. One of his more often-quoted pithy sayings is "But interior decorating is a rock hard science compared to psychology practiced by amateurs." He wrote this with regard to the Supreme Court trying to determine whether religious practitioners' beliefs were legitimate or not - Justice Scalia (then) took the position that such was an improper inquiry for the court. Potentially up until the Hobby Lobby case, that view held sway.

3. For my money, the best option at least for pith is Alex Kozinski of the 9th Circuit Court of Appeals. Some of his better dissents:

a. this dissent in a criminal appeal, which did not win the case, but did, remarkably, convince the US DOJ to release the prisoner. It is well worth reading in full.

b. White v. Samsung, where he dissented against a ruling creating a right to publicity when Samsung's advertisement featured a robotic Vanna White. Ms. White sued because she felt she should be paid and won over Judge Kozinski's dissent.

c. US v Alvarez, where Judge Kozinski expressed his certainty that the Stolen Valor Act is unconstitutional, because there is a First Amendment right to be untruthful. Again, well worth reading in full.

d. Redondo Beach, where Judge Kozinski wrote a fairly blistering dissent when the majority opinion said there was a right for day-laborers to congregate on street corners.

I'm not sure who has the most SCOTUS dissents, but it's worth mentioning that frequent dissents are a somewhat recent innovation. Before Justice John Marshall took over as chief, all the justices published separate (seriatim) opinions. Afterward, the norm became unanimous decisions, with only very occasional dissents. Over time dissents have become more common, although most cases are still unanimous.

Also worth mentioning that concurrences are often as important as dissents.

Justice Scalia is perhaps the best known dissenter of the last several decades although this is probably due much more to the high quality and entertainment value of his dissents rather than a high quantity.

Apples to apples is hard, simply because the court has been steadily hearing more and more cases, but I think that we're remiss if we don't mention William Douglas.

The rumor is that Douglas was audited by the IRS, or that he had a particularly prickly time with them at some point, but there's nothing to support that. It seems as though he turned into a one-man crusade against the tax code (though his voting record is a little more complicated than it's often made out to be, particularly over the course of his career, like when he was the lone dissenter for the government and not the taxpayer). Never one for subtly, his tax dissents are sloppy, when he cares to write at all (he would often dissent without opinion, which, while relatively common in the early court, started to become pretty rare around the time that he joined the court).

Also, in famous dissenters, we need to add the PA Supreme's Michael Musmanno to the list. I'll put in the edit for his that reads, if memory serves, like the end of Ulysses but for negation, but one of his more fabled ones - and you really have to read it, to quote it is to almost do it an injustice, is Commonwealth v. Robin:

{Henry Miller's Tropic of Cancer} is not a book. It is malignancy itself. It is a cancer on the literary body of America. I wonder that it can remain stationary on the bookshelf. One would expect it to generate self-locomotion just as one sees a moldy, maggoty rock move because of the creepy, crawling creatures underneath it.

I'd like to know who wrote the most separate opinions (average per term), with no other justices joining the separate opinion.