I had a Sega Megadrive/Genesis, it was the most fun! I literally spent hours playing it every day when I was a kid. By the late 90s I had amassed around 40 games, most of which I sold at some point. The Sonic series was epic and Streets of Rage also. But since Bobby already posted those I'll post a few more of my faves.
Toejam & Earl:
Epically awesome, especially in split-screen multiplayer mode. You play two aliens who crashlanded on earth. You have to collect pieces of their rocketship over 25 levels while dealing with all sorts of random human enemies, like the crazy dentist, Cupid, dancing hula girls, the Boogieman, the evil Mole, and more. You collect presents that are scattered around that give you special powers like Rocket Skates, Icarus Wings, Warp Doors, Bouncing Shoes, and more. But there's bad presents too that make you lose life, or randomize the presents you have so you don't know what they are. Fucking epic.
Toejam & Earl 2: Panic on Funkotron
Possibly even more fun than the original, but a completely different type of game, this was a side scrolling platformer. On this one, the two funky aliens are back on their home planet of Funkotron, but accidentally brought a bunch of humans back with them, so you have to travel around their planet collecting humans in Jars and shipping them back to Earth. You have to solve all sorts of puzzles along the way like collecting all the favorite possessions of Lamont the Funkopotamus who is hiding in the Funk Dimension due to being scared of the humans. This game is so awesome. I just heard that it's available on the Wii Virtual Console so when I get back to the UK I'm going to buy it on there.
Michael Jackson's Moonwalker:
My second favorite video game of all time. What's not to love about a game where you play as Michael Jackson, moonwalking, dancing, throwing your hat at bad guys, and occasionally transforming into the killer MJ robot from the movie, all while traversing through levels vaguely resembling Mike's music videos, set to MIDI versions of "Smooth Criminal", "Bad", "Another Part Of Me" and "Billie Jean"? This game is beyond epic.
Kid Chameleon:
This game was ridiculously hard but so much fun. It had something like 60 levels and maybe more, and like most games at that time, no save function. I remember leaving my Megadrive paused and turned on overnight just to continue where I left off. This game had you controlling a kid stuck inside a virtual reality arcade game. You collected different helmets that gave you different powers, like a ninja/samurai helmet, one that turned you into a fly so that you could fly to higher parts of the level, and one that turned you into a Jason Voorhees type character with a hockey mask that could throw axes at enemies. Stupidly fun.
Decap Attack
This game made absolutely no sense but it was super fun. Apparently there was a Japanese game called Magical Flying Hat Turbo Adventure based on an anime TV show, but the license was only for Japan, so the developers created all new levels, all new music, all new character, an all new story line based on the original game engine, and it became Decap Attack internationally. Ridiculous game but really fun.
I'll post more later.