http://www.opensourceshakespeare.org/views/plays/characters/chardisplay.php?sortby=lines&searchterm=

UPDATE – OpenSource Shakespeare has actually removed the “sort by lines” feature and replaced it with “sort by speeches”. This is somewhat more accurate, as there is no set rule about line breaks in most of the text.
However, counting speeches isn’t remotely as fun. One character might have 10 small speeches, compared to 3 lengthy speeches from someone else.
The results are still what you’re thinking – Falstaff, Richard III and Henry V top the list, all of whom appear in multiple plays. The remainder of the top ten might as well be, “Title Characters” — Hamlet, Othello, Antony, Cleopatra. Even Timon (of Athens) gets in there.
The one spoiler? Iago, He’s got almost as many speeches as Othello!
Open Source Shakespeare is a great resource for doing things like this (not my idea, I just found it). Here, we have characters sorted by the number of lines. Obviously, it’s a little crude as Shakespeare himself shows up in the #1 spot with no plays listed, but right behind him are Falstaff and Henry V (both having appeared in numerous plays with major roles, it only makes sense), and then Hamlet (who, for only have a single play, has the most lines). Other interesting bits…
Othello and Iago have almost an identical number of lines.Anthony has more lines than Cleopatra – but he was in Julius Caesar, too.Behind Cleopatra, Rosalind from As You Like It has the most lines for a female.Romeo has substantially more lines than Juliet, though they both have quite a few. Lady Macbeth, on the other hand, has barely 1/3rd the number of her husband’s lines. Tybalt barely registers with just 17 lines!