A frequent question which occurs on the newsgroup is: What Renaissance choral masterpieces use large numbers of independent vocal lines?
By far the most famous of these is the 40-part "Spem in alium" of the English composer Thomas Tallis (c.1505-1585).
Other prominent examples include the 36-part canon "Deo gratias" sometimes attributed to Johannes Ockeghem, the 19-part "O bone Jesu" of Robert Carver (c.1484-c.1568), and the 12-part "Earthquake Mass" of Antoine Brumel (c.1460-1515). Of course, there are also many other examples (especially from Italy c.1600), but they are generally less well-known individually.
If you are looking for recordings, here are a few. The first is especially aimed at illustrating this style of piece.