Your question is a tricky one.

Originally, the pope was simply the leader (bishop) of the church in Rome. (Officially he still is, but today's popes hardly concern themselves much with leading their local church.) So while it may be well established that some fellow led the church in Rome some time in the second or third century, whether he acted as the Pope is not such a clear-cut thing. There are references in second-century sources to the pimacy of the Bishop of Rome before all other bishops, and this primacy was noted by the First Council of Nicaea (in 325), but the amount of actual influence and practical authority that the Bishop of Rome held over christians is a complicated issue.

So, let us leave the "pope" question aside, and simply ask who was the first historically verifiable Bishop of Rome. That's an easier question, but it's still a little slippery.

It seems to be generally agreed among historians that all those on the official Catholic list of popes were quite likely real people (although reliable sources for Anacletus, 3rd on the Catholic list, are few), and that all of them -- with the possible exception of Peter -- held some leadership position among the christians in Rome.

But records of the early churches are fragmentary. Remember that Christianity was illegal in the Roman Empire until the Edict of Toleration (in 311) and the Edict of Milan (in 313). The early churches didn't exactly publish their records for all to see. Our sources about the early church are often letters or essays that mention people in passing, or histories written years later.

And our sources disagree. There is inconsistency in the order in which leaders were said to have held their offices, and in which years they did so.

For example, there are early sources listing Linus as the second Bishop of Rome -- or the first, if they didn't count Peter. Irenaeus, writing not long before the year 200, said that Linus followed Peter. OTOH, Tertullian, writing not long after 200, wrote that the successor of Peter was Clement (i.e., Clement I), while Catholic tradition places Clement fourth.

But as the years go by, things become clearer. Serious disagreements in our sources largely die down around Zephrinus (Bishop of Rome 199-217, 15th in the Catholic list). Starting with Pontian (Bishop of Rome 230-235, 18th on the Catholic list), we have not only years, but generally agreed upon dates for the beginning and end of the tenure of each Bishop of Rome.

And then during the tenure of Miltiades (Bishop of Rome 311-314, 32nd on the Catholic list), Christianity became legal. In 380, during the tenure of Damasus I (pope 366-384, 37th on the Catholic list), Christianity became the state religion of the Roman Empire. The records are quite good from that point on.

To sum up, the answer to your question depends a great deal on what exactly you mean by it. If you want a quick answer, I'd have to say Clement I (Bishop of Rome in the years just before 100 -- probably -- and 4th on the Catholic list). But if your standards for "well documented" are higher than that, then you want someone rather later. And for someone who truly acted as the leader of christians worldwide, you'll have to go centuries later.

While there is some extant writing apparently concerning Pope Anacletus (or Cletus), who directly succeeded Linus, this cannot be independently verified and there is precious little of it, so let's call his existence "doubtful" for historical purposes.

The next in line would be Pope Clement I, who is also, probably, our best candidate for "earliest confirmed Pope," both by Christian sources and (possibly) independent sources at the time and in the region. His writing is known in the Roman Catholic Church by his first epistle, and certainly establishes him as having existed (much scholarship has been done on his epistles, which confirms this was written by him, even as placing into serious doubt others which had been ascribed to him) and having had the role of Bishop of Rome (or Pope). Interestingly, he is the only Roman Catholic Saint to have an Eastern Orthodox church named after him -- further demonstrating his importance and existence in Christianity.

However, Clement I, like several of his immediate successors and predecessors has more legend and mythology surrounding him than actual biographical data. If we want to absolutely, positively, for sure know that our early Pope existed and held the title, our first, best candidate would be either Pope Eleuterus or, at the very latest, Pope Callixtus I.

Eleuterus was discussed independently from strictly Christian sources by his contemporary Saint Hegesippus, a Christian convert from Judaism. Hegesippus, however, had made a large study of the lives of previous Christian saints, etc., and could possibly -- although very unlikely -- have been making certain things up about Eleuterus, so it's slightly possible we can't trust his stories. If we really want to be ironclad, we can discuss Callixtus I, for whom writings about him by his enemies exist, including the Roman writer Tertullian, who mentions Callixtus by name and title. This would most certainly qualify him as legitimate.

Both Eleuterus and Callixtus I lived and died around the end of the 2nd and beginning of the 3rd centuries A.D.