I've beaten all the Super Mario Bros games, up until the new gen of consoles. (e.i 3DS, Wii-U). All the Mario RPGs, Most of the mario parties, All Mario karts and Tennis/golf ones. (The one of GBA is awesome)

I'll just keep it strictly Super Mario platformers (both 2D and 3D), in which case I've beaten them all.

edit: nevermind, timestamp on youtube video i posted didn't work... joke broken. 

Super Mario Bros
Super Mario Bros 2
Super Mario Bros 3 (best Mario ever)
Super Mario World
Super Mario Land 2 (Gameboy)

After that I pretty much lost interest in Mario. Never liked the move to 3D. But to this day I still have my original working NES and sometimes play drinking marathons of Super Mario 3 with some buddies. Good times.

Super Mario Bros.
Super Mario Bros. Deluxe (including all bonus levels)
New Super Mario Bros.

Super Mario 64
Super Mario Sunshine
Super Mario 64 DS
Super Mario Galaxy
Super Mario Galaxy 2
Super Mario 3D Land

Mario Kart 64
Mario Kart: Super Circuit
Mario Kart: Double Dash!!
Mario Kart DS
Mario Kart Wii
Mario Kart 7

Mario Tennis
Mario Golf: Toadstool Tour
Mario Power Tennis
Mario Tennis Open

Mario Party 1
Mario Party 2
Mario Party 3
Mario Party 4
Mario Party 5
Mario Party DS

Paper Mario

Luigi's Mansion (?)

Yes!

Another who sees through the facade of Super Mario World and realizes Yoshi's Island rules all !

This one?

Rare to see someone having beaten Hotel Mario, a licensed but ignored by Nintendo CDi game usually considered to be extremely bad especially due to the ridiculous cutscenes, and not one of the 'real' Mario games.

2D Mario:

Super Mario Bros. (NES)
Super Mario Land (GB)
Super Mario Land 2: Six Golden Coins (GB)
Super Mario Bros. Deluxe (GB)
Super Mario World (SNES)
Super Mario World 2: Yoshi's Island (SNES)
Super Mario Advance (GBA)
Super Mario Advance 3: Yoshi's Island (GBA)
Super Mario Advance 4: Super Mario Bros. 3 (GBA)
Super Mario All-Stars: 25th Anniversary Edition (Wii)
New Super Mario Bros. (DS)
Yoshi's Island DS (DS)
New Super Mario Bros. Wii (Wii)
New Super Mario Bros. 2 (3DS)
New Super Mario Bros. U (Wii U)

3D Mario:

Super Mario 64 (N64)
Super Mario Sunshine (GCN)
Super Mario 64 DS (DS)
Super Mario Galaxy (Wii)
Super Mario Galaxy 2 (Wii)
Super Mario 3D Land (3DS)

Mario Kart:

Mario Kart 64 (N64)
Mario Kart: Super Circuit (GBA)
Mario Kart: Double Dash!! (GCN)
Mario Kart DS (DS)
Mario Kart Wii (Wii)
Mario Kart 7 (3DS)

Mario Party:

Mario Party (N64)
Mario Part 6 (GCN)

Mario Sports:

Mario Tennis (N64)
Mario Power Tennis (GCN)
Mario SuperStar Baseball (GCN)

Mario RPG:

Paper Mario (N64)
Paper Mario: The Thousand Year Door (GCN)
Super Paper Mario (Wii)
Paper Mario: Sticker Star (3DS)
Mario & Luigi: Superstar Saga (GBA)
Mario & Luigi: Partners in Time (DS)
Mario & Luigi: Bowser's Inside Story (DS)

Mario Miscellaneous:

Mario Pinball Land (GBA)
Mario vs. Donkey Kong (GBA)

Total:

41 Mario Games

From what I can remember:
3DS: Super Mario 3D Land, Mario Kart 7.
Wii: Mario Kart Wii, Super Mario Galaxy.
GameCube: Mario Kart Double Dash!!

That's it. I have had quite a lot of Mario games but I never got around to finishing them.

Yeah! :) Well actually I don't mean to disrespect ANY of the old school Marios, each of them completely fascinated me at some point. But Yoshi's Island is especially vivid in my mind for feeling super inspired and silly, in the best way.

SMB still feels super slick today. Crazy how they nailed it with their first try. 
SMB2, my favourite before Yoshi's Island came along! Yeah I know, it's the ugly duckling, but I loved the atmosphere and strangeness.
SMB3 is the one I like the least today, mostly for the physics - it's a bit slower than the rest. 
SMW, haven't replayed it since the actual SNES days. But it was the best thing back then.

Regarding the "New" series, I know they get a lot of criticism, but I think they successfully evolve the pure gameplay, meaning how Mario feels and bounces. Still best in the genre. They don't innovate enough in the atmosphere / crazyness department, though.