Here's some of my requests: (some of which I mentioned before) It will be awesome if the complete run is done without any continues or failures.
Ski Champ (SEGA ski racing game): do two runs that show that layout of each course at least once.
Code One Dispatch (Konami racing game)
Rambo (2009 Arcade game)
F-1 Grand Prix Star: Do all the courses
F-1 Grand Prix Star 2: Do all the courses
Aqua Rush (a Namco arcade puzzle): complete at least Hard mode. [optional: complete all three modes]
Star Sweep (a puzzle game)
Capcom Sports Club (a Capcom set of three sports games: Tennis, Basketball, and Soccer): complete run on all three sports. [according to a website, you can play against secret teams on each sport by getting enough points before winning the finals.]
Magical Truck Adventure (arcade): show several runs that show all six possible courses
Tenkomori Shooting (Namco arcade game): complete all three courses
Tetris Plus 2 (arcade puzzle): show all puzzles and endings
Virtua Cop 3 (SEGA shooter arcade)

Optional Games:
LA Machineguns (Arcade)
The Ocean Hunter (a SEGA shooter)
Star Wars Trilogy (Sega arcade game)
Slipstream: Do World Championship (show all four courses) and Passing Contest
Club Kart European Session: do all the courses
JoJo's Bizzard Adventure (PS): complete "Super Story Mode" with a S Rating (if possible)
The Maze of the Kings (SEGA shooter): grab every artifact.
Rhythm Thief & the Emperor's Treasure (3DS): get A on every chapter