Which player has the most Champions Leagues?

Short answer: Francisco Gento (Real Madrid, 6 European Cups in the 1950s-60s) holds the record.

Detailed answer

Real Madrid's Francisco Gento won 6 European Cups (1956-1966) — pre-Champions League era but counts. Modern era: Cristiano Ronaldo (5 CLs), Toni Kroos (6 with Bayern + Real). Among TenDollarPick players: Cristiano Ronaldo has 5.