The Video Game Critic

Oh, let's just get this out of the way already.

Rashied Ali
Birdstuff
Neil Peart
Bun E. Carlos
Billy Cobham

I love these music lists. In no particular order:
Terry Bozzio (UK, Frank Zappa) - Not really a Frank Zappa fan but I found this performance amazing https://www.youtube.com/watch?v=R2gLggbIipc
Phil Collins (Genesis) - Probably the most underrated drummer in my opinion. In the heyday of Genesis, most people didn't even realize he played drums.
Carl Palmer (ELP, Asia)
Alan White (Yes)
Roger Taylor (Queen)
Stewart Copeland (Police)
Bill Bruford (King Crimson, Yes)
Simon Phillips (Various)
Pat Mastelotto (King Crimson)
Cozy Powell (Jeff Beck, Rainbow, solo, various)

Other than Cozy Powell and Roger Taylor, mostly progressive music

Lots of great picks so far - loving the prog and fusion drummers I'm seeing:

Dave Lombardo (Slayer)
Brann Dailor (Mastodon, Today Is the Day)
Gene Hoglan (Strapping Young Lad, Testament, Death, Dethklok, etc.)
Danny Carey (Tool)
Buddy Rich
Gene Krupa
Tomas Haake (Meshuggah)
Jimmy Chamberlain (Smashing Pumpkins)
Max Roach
John Bonham (Led Zeppelin)
Keith Moon (The Who)
Matt Cameron (Soundgarden)
Dale Crover (Melvins)
Richard Hoak (Brutal Truth)
Pete Sandoval (Morbid Angel)
Mick Harris (Napalm Death, Painkiller)
Jan Axel Blomberg (literally dozens of black metal bands)
Ben Koller (Converge)

Ringo Starr

Danny Carey (Tool)
Jimmy Chamberlin (Smashing Pumpkins)
Matt Cameron (Soundgarden)
Abe Cunningham (Deftones
John Dolmayan (System of a Down)
Tim Alexander (Primus)
Mike Bordin (Faith No More)
Phillip Selway (Radiohead)
Bud Gaugh (Sublime)
Chad Sexton (311)
Dave Grohl (Nirvana)

1-Neil Peart

2-Ticco Torres

3-Rick Allen

4-Deen Castranovo

5-Chris Whipper Layton

6-Matt Sorum

7-Rob Reiner.

8-Mick Brown

9-Frank Beard

10-Lars Ulrich