Mario, Metroid, 2D Zeldas including A Link between Worlds, Mother, Donkey Kong: Everyone fails to get how fun, imaginative and challenging Nintendo games are, and that is a shame, because they are rarely boring. All of these series are awesome and I have been playing them my whole life without getting bored.
Bionic Commando: A game that everyone should play, as the unique switch from jumping to grappling makes it special and wondrous.
Mega Man 1 through 5, 9, 10, X1 and 2: Classic games, and recently given a great retro rekindling, in both the senses of restarting and fizzling out quickly. It may be a dead series, but I still love it.
Doom, Rise of the Triad, Duke Nukem 3D, Quake, Quarantine: I long for the days when I could distinguish between First Person Shooters, and these are the perfect examples: Their fast-paced action, complex levels and nice enemies really stnd in stark constrast to the brown corridors and generic humans of modern shooters.
Lemmings: Classic puzzle game with lots of levels, humor, bloodthirsty slaughter and style. I will never get tired of watching Lemmings et crushed.
Popful Mail: Yes the Sega CD had a classic game: A stylish, imaginative action-R.P.G. with lots of great characters and interesting levels. Well translated with a sense of humor and a higher difficulty level.
Tetris: Definitive, gaming boiled down to the simplest possible task yet infinitinely replayable. Never feels repetitive, just a perfect way to spend time. Go to Tetris Friends and you can play a version where at level 20, all the pieces become invisible: Tough at first, but eventually, you learn to place blocks instinctively, rather than consciously, and it becomes reasonable.
Bayonetta: Constant over-the-top action and weirdness mixed with strange characters and elaborate levels. Most of the worst parts of the first, like button mashing and Quick Time Events, were removed from the sequels, making it perfect.
Sierra, Lucasarts, Legend and Infocom games: Hilarious games with clever puzzles and great characters. There are some shitty ones like Phanasmagoria, Quest for Glory 2, Companions of Xanth, Gabriel Knight 2 and 3, Plundered Hearts and Spellcasting 201 but they are the minority.
Vice City: Best of the series by far, with an awesome '80s style, classic music, hilarious jokes, great mission design and a real sense of progression. I replay this every few years. Fuck the censors for hating this series. Everyone should play this, and ignore whatever arbitrary age rating is on it.
Contra, Bucky O'Hare, Castlevania, Ai Senshi Nicole, Sunset Riders, Parodius, Twinbee: Konami's definitive NES and SNES action games, with lts of fast-paced action, imagination, and challenge. Unfortunately, the only great Konami series these days is Metal Gear Solid, and the company seems to be going out of business. That Castlevania God of War clone was a worthless piece of shit, though still far superior to God of War.
Little Nemo, Ducktales, Chip and Dale, Gun Smoke, Devil May Cry, Resident Evil 4, 5 and Revelations, Dragon's Dogma, Ghosts and Goblins, Street Fighter 2 and 4: Capcom's action classics, and even now, despite all the troubles, it still sometimes produces classics.