Which city's library rivaled Alexandria, built at Pergamon?

Pergamon itself, in what is now western Turkey, whose library was the serious competitor to Alexandria and reportedly held around 200,000 scrolls.

The rivalry produced parchment. Egypt, the story goes, cut off papyrus exports to strangle the competition, so Pergamon developed treated animal skin instead - the Latin pergamena and the English parchment both carry the city's name. Whether the embargo happened is doubtful and skins had been written on long before, but Pergamon does seem to have refined the process, and the consequence was enormous: parchment can be written on both sides and folded, which makes the bound codex possible and eventually kills the scroll.

The collection is said to have been given to Cleopatra by Mark Antony as a wedding present, which would mean the loser's books ended up in Alexandria after all - though the source for that is hostile and probably making a point about Antony giving Roman property to a foreign queen. Pergamon's other export was medical: Galen trained there, treating gladiators at its temple of Asclepius, and his anatomy governed European medicine for fourteen centuries despite being based largely on dissected pigs and monkeys.