Just a heads up, with me, this'll be mostly beat 'em ups.
Armored Warriors - Another Capcom beat 'em up. Kind of a take on Cyborg Justice, but with giant mechs, and easier controls.
Asterix - European Konami beat 'em up, centered on picking up enemies(what direction you press when you do determines your throw). Also kind of difficult, I think you can actually game over, regardless of coins.
Battle Circuit - Another Capcom beat 'em up, of the wackier variety.
Battletoads - Very different from the console versions(gory too).
Bubble Memories: The Story of Bubble Bobble III - only arcade exclusive Bubble Bobble game.
Bucky O'Hare - Konami beat 'em up/run 'n gun hybrid.
Denjin Makai - Beat 'em up, one of the best. This is kinda tough because it does have a half ass port on the Super Famicom(which is still one of the better ones on the console) called "Ghost Chaser Densei."
Golden Axe: The Revenge of Death Adder - Only arcade exclusive Golden Axe.
Guardians/Denjin Makai II - Probably the greatest arcade exclusive beat 'em up.
Dungeon Magic - Another beat 'em up, isometric view, kind of like the lovechild of Landstalker & Streets of Rage. It is on a compilation for the PS2 though.
Mega Man: The Power Battles; & Mega Man 2: The Power Fighters - Basically Mega Man, if they mashed some of the games together, and eliminated everything but the boss battles.
Metamorphic Force - Konami beat 'em up that uses the X-Men engine, and introduces were-creatures as playable characters.
Muscle Bomber Duo: Ultimate Team Battle - Tag team version of Saturday Night Slam Masters.
Night Slashers - Horror beat 'em up, one of the best arcade exclusives.
Nitro Ball - Run 'n gun & pinball combination.
Osman - Arabian themed Strider clone.
Riding Fight - beat 'em up that takes place mostly on hover boards. Has an effect similar to SNES' mode 7.
Ring of Destruction: Slam Master II - European fighting game sequel(as in no pinning/submission this time) to Saturday Night Slam Masters
Rollergames - Completely different than the NES game, it's like a beat 'em up sport.
Shadow force - 6 button Technos arcade exclusive beat 'em up, that involves playing as ninja who can possess the bodies of their enemies.
Snow Bros. 2 - The original got ported to the NES, Game boy and Genesis, but this remains arcade exclusive.
Spider-Man: The Video Game - Beat 'em up by Sega.
Super Dodgeball/Kunio no Nekketsu Toukyuu Densetsu - Probably the only arcade exclusive Nekketsu/Kunio game, large sprites, fighting game style super moves, optional defeated character recruitment, it's pretty good.
The Simpsons; & X-Men - Unless you're counting X-Box Live. Konami beat 'em ups.
Undercover Cops - Another beat 'em up, and again, it did have a Super Famicom port. If you go for this, be sure to also get, and only ever use, the "Alpha Renewal Version." The worldwide release was basically an unfinished demo(a lot of moves missing among other issues) compared to the Japanese release, so "Alpha Renewal" fixes that
Vendetta; and Violent Storm - XYXZYZ brought up Crime Fighters, these are it's sequels(Crime Fighters, Vendetta, Violent Storm, in that order). Violent Storm plays like a Capcom beat 'em up, but adds jump attack cancels into other jump attacks, and hitting enemies while they're on the ground(which is a staple of the Crime Fighters trilogy).
Warrior Blade: Rastan Saga Episode III - The only arcade exclusive Rastan game, this Japan only release is similar to the Golden Axe series, but has a more impressive move set, and unique random moments between stages that can bring random luck(good or bad).
Warriors of Fate - Capcom beat 'em up, in the 3 Kingdoms, but with names changed for the U.S. release. Has a Saturn version, but that never left Japan.
Wild West C.O.W. Boys of Moo Mesa - Similar kind of run 'n gun to Sunset Riders.
WWF Wrestlefest - Probably the best licensed arcade wrestling game.