As you know, the Conclave of Cardinals meets today to consider who will be the next pope.
Some interesting near-popes:
* Valentinus (c. 100 - c. 160): Valentinus, a Gnostic heretic, is at least an alleged near-pope; if I recall correctly, Tertullian writes somewhere that he had almost been elected bishop of Rome, at some point before he fell into heresy.
* Pedro de Luna (1328-1402): During the 1378 conclave, he was the Italian who was most likely to be Pope. Pedro de Luna voted for the man who was elected Pope Urban VI, but later was convinced that the election was invalid, and helped to elect and support the first anti-pope of the Avignon schism, Clement VII. He was elected the second Avignon pope: Benedict XIII.
* Johannes Bessarion (c. 1400-1472): Bessarion was one of the greats of the Italian Renaissance. One of the Greek bishops who was involved in the Council of Florence, he was made cardinal by Pope Eugenius IV.
* Hyacinthe Sigismond Gerdil (1718-1802): In the seventeenth century, certain Catholic nations claimed that they had a right to exclude a papal candidate (the veto could be used only once during the conclave). Gerdil, the last great Malebranchean, failed to be elected due to such a veto; the Germans didn't want a Frenchman on the papal throne. His best-known work is a defense of Malebranche against Locke. Needless to say, Gerdil is my favorite almost-pope.
* Mariano Rampolla (1843-1913): He was also vetoed, by the Austrians. The person who became pope instead, Pius X, ruled that any cardinal who put forward such an exclusion in papal elections in the future would be excommunicated. It was probably never stricly legal in the first place; who knows what a Gerdil or Imperiali or Rampolla papacy might have been if people had not simply gone along with it?