Picture : the 3 players of the Soviet Union National Team : Yashin, Ivanov and Ponedelnik.

Players with most finals: 22 men

Three players of the Soviet Union national team - Valentin Ivanov, Lev Yashin and Viktor Ponedelnik - took part in the final games of the first two European Cup editions (1960 and 1964). 
They still hold the record for the number of final participations, but over the years as many as 19 more players joined them: 10 from Germany, 8 from Spain and 1 from Portugal. 
So, the list of record holders is now exorbitantly inflated. But in the upcoming tournament, there is a good chance that the old record will finally advance.
Sergio Ramos and Cristiano Ronaldo, two from this long list, will surely take part in the tournament. If Spain or Portugal (or both teams) make it to the final, we will celebrate the first three-time European Championships finalist(s).
Each of the 22 record co-holders has won at least one of their finals (winning years are taken in bold in the table below), and 8 Spaniards have done both theirs.
All recordsmen but Cristiano Ronaldo participated in the finals in two consecutive tournaments. The Portuguese achieved this with a spread of 12 years (2004 and 2016).

Valentin IvanovSoviet Union21960, 1964
Lev YaschinSoviet Union21960, 1964
Viktor Ponedelnik Soviet Union21960, 1964
Franz Beckenbauer Germany21972, 1976
Uli HoeneßGermany21972, 1976
Sepp MaierGermany21972, 1976
Georg SchwarzenbeckGermany21972, 1976
Herbert Wimmer Germany21972, 1976
Bernhard Dietz Germany21976, 1980
Thomas HäßlerGermany21992, 1996
Thomas HelmerGermany21992, 1996
Jürgen KlinsmannGermany21992, 1996
Matthias Sammer Germany21992, 1996
Xabi AlonsoSpain22008, 2012
Iker CasillasSpain22008, 2012
Cesc FabregasSpain22008, 2012
Andres IniestaSpain22008, 2012
Sergio RamosSpain22008, 2012
David SilvaSpain22008, 2012
Fernando TorresSpain22008, 2012
XaviSpain22008, 2012
Cristiano Ronaldo Portugal22004, 2016