To quote Highlander : "There can be only one!"
Loved that. I remember playing the Spectrum version with two players. Many hours ploughed into Bub and Bob there. I was a bit upset that the sequel(Rainbow Islands) only allowed alternate turn play for two players. Still at least Parasol Stars reverted back to Bubble Bobble's game mechanic.
As for other games :-
Pitstop II on the C64 - Knackering your opponents tyres was a dirty way of driving - worthy of Schumacher.
Populous I and II on the Amiga - via serial link-up. Great stuff here. Could be a lengthy game though. Loved using the different powers in the second game and the knights as well. Also, watch out for "the plague"!!!!
Stunt Car Racer on the Amiga - via serial link-up. Turn around and try to ram the other guy. Hehehehehehehehe! great stuff.
Knights of the Sky on the Amiga -via serial link-up. Dogfighting in WWI. Great stuff. One trick was to land and hide for a while, then take off and attack!
Lotus I and II on the Amiga. i think the first game was slightly better as this allowed for racing for positions. The second game could allow for serial link-up and also 4 players. Amazing stuff.
Chaos Engine on the Amiga. Completed it this way. Loved the characters on this.
SWIV on the Amiga. Another game completed this way. I feel sorry for the poor person who is the tank.
Battle Isle '93 - Turn based game. Think of it like Chess except with futuristic vehicles.
Historyline 1914-1918 - Another turned based game from Blue Byte(see above). This time around it's based on WWI. You even get a train as vehicle.
Final Fight at the arcade. Great fun with two players.
Mercs at the arcade. Only played it with two players - so missed out on the three players version.
Smash TV at the arcade. Crazy at times, but fun.
SF 2 at the arcade. Beating your opponent was fun.
Metal Slug 2 at the arcade. Good shoot 'em up fun..
Doom 2 on the PC via serial link up and also at one time modem. Cool stuff. Loved the double barreled shotgun in that. Get in close and BOOM!!!
Duke Nukem on the PC via serial link up and modem. Hehehe. So much fun with the maps and so on. You could lay those laser trip bombs in a sneaky place and watch the other guy explode. The shrinking ray was funny. Just remember to tread on the guy afterwards. The freeze gun was good as well - but remember to smash the opponent afterwards.