I will make one thing clear from the get go: this question is asked from the perspective of an atheist. And based on my world view, I am pretty certain of two things:
Pope Francis is very much a real person
The existence of the first pope, St Peter, is debatable
However; there is an alleged unbroken chain of 264 other popes, connecting the two men through history. So my question is this: Who is the earliest "historical" pope, meaning the first pope that we know was an historical figure?
It's an older thread and the one who answered the question appears to have deleted his account, but you might look here while we await a reply.
AttilaTheFun818's link deals with the issue of what pope meant in the early church so lets look at existance.
Peter existed. Paul of Tarsus mentions him in the letter to the Galatians.
After that well somebody wrote the letter of Clement to Corinthians and its entirely consistent with being written by someone senior in rome during the right time period. Clement also appears in the early histories.
Between the two (or not early histories go Peter to Clement or only have one person between the two) you have Linus and Anacletus that there is rather less evidence for. Dosn't mean they didn't exist and its possible the confusion is due to the idea of a single top guy in rome not existing at that point.
The next few popes are rather questionable. There was a christian community in rome during the period and it did contain some senior people but matching the names to actual people is iffy.
Depending on how you view Eusebius of Caesarea then the early second century Telesphorus is the next pope with a historical record (in the context of the dispute over when to celibrate Easter). If you want to avoid Eusebius then the mid second century Pius I where we have multiple near contempary accounts of the guy existing (well recently being dead anyway which strongly suggests he existed). Pius is also the point where the Liberian Catalogue goes from simply listing dates to actualy saying something about the pope in question (although it gets the order slightly wrong).