Who Were the Earliest Popes?

With the news of Pope Benedict's resignation at the end of February, it seemed like a good time to recall the names of our earliest popes.  There's this idea out there that the papacy didn't really come together until the 400's, and that's flat out wrong.  Irenaeus, the Bishop of Lyons (France), in his Against the Heresies, put forward the Bishops of Rome from Peter to the time of the work’s composition (c.189 A.D.):

Since it is too long to enumerate in such a volume as this the succession of all the Churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion assemble other than where it is proper, by pointing out here the successions of the bishops of the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious Apostles, Peter and Paul, that Church which has the tradition of the faith which comes down to us after having been announced to men by the Apostles. For with this Church, because of its superior origin, all churches must agree, that is, all the faithful in the whole of the world; and it is in her that the faithful everywhere have maintained the Apostolic tradition.

The Blessed Apostles [Peter and Paul] having founded and built up the church [of Rome], they handed over the office of the episcopate to Linus. Paul makes mention of this Linus in the Epistle to Timothy. To him succeeded Anencletus; and after him, in the third place from the Apostles, Clement was chosen for the episcopate. He had seen the Blessed Apostles and was acquainted with them. It might be said that he still heard the echoes of the preaching of the Apostles, and had their traditions before his eyes. And not only he, for there were many still remaining who had been instructed by the Apostles.

. . .To this Clement, Evaristus succeeded; and Alexander succeeded Evaristus. Then, sixth after the Apostles, Sixtus was appointed; after him, Telesphorus, who also was gloriously martyred. Then Hyginus; after him, Pius; and after him, Anicetus. Soter succeeded Anicetus, and now, in the twelfth place after the Apostles, the lot of the episcopate has fallen to Eleutherus. In this order, and by the teaching of the Apostles handed down in the Church, the preaching of the truth has come down to us (Book III, 3:2-3).

So there you go.  Who will be the next man to act as Christ's Prime Minister?  I can't wait to see!