It's fun to get to level 50 on Galaga or play Pac-Man for an hour, but not if you are playing two players... (unless the 2nd player needs some sleep or likes to read). What's more fun is two player games where you go mano a mano with the other player. Not cooperatively shooting, or taking turns, but where your goal is to destroy the other player and have your victory reduce them to a sobbing sack of goo.
So here are the games that have already been Games of the Day that are especially fun because of the two player competition. Some of them are VERY simple games but tons of fun nonetheless. Obviously, EVERY good Vs. fighting game would fall into this hall of fame, but remember this is only for games that have already been GOTDs and I try to make those more obscure games, so you won't see any Street Fighters or Mortal Kombats here.
1. Change Air Blade
The only vertical space shooter I know that is 2 player versus.
Classic breakout style fun versus other players and the CPU.
3. Title Fight
Split Screen boxing with dual joystick controls.
4. Hyper Athlete
A modern descendent of Track & Field with some very long events.
5. Burglar X
Zany modern maze shooter - race the other player to collect coins - winner play alone.
6. Off the Wall
Pong meets Breakout with lots of ingenuity.
Split-screen Cross-Hair Shooter - you shoot at each other buildings.
8. B.C. Story
Caveman Track & Field with witty use of controls and goofy cartoons.
Simple racquetbally goodness.
10. Hat Trick
One on One (plus goalie) hockey fun.
11. Block Hole
It's kind of a split screen tetris-y shooter. Very addictive.
Classic puzzle game - funner with 2 folks.
13. Star Guards
Cool controls - old schoolish - Winner of each round gets bonuses and power-ups.
14. Heavy Smash
Updated version of Speedball - kinda like football meets hockey meets rugby.
15. Bang Bead
A modern twist on Pong. Two opposite players whip the ball at each other.
Early Split-screen Doom-type game - Hardest I ever laughed playing a game.
Precursor to Bloody Roar - Change into your animal and fight.
18. Plasma Sword
Cool 3D fighting with lots of characters.
19. Super Gem Fighter Mini-Mix
Child-Mutant versions of classic fighters and many cool twists.