Gradius 2
Salamander
Contra
Super C
AVOID CONTRA FORCE
Crystalis
Startropics

Super Mario Bros. 3
Super Mario Bros. 2
Faxanadu
Final Fantasy
Mega Man II
Mega Man III
Castlevania III
Kirby's Adventure
River City Ransom
Over Horizon
Crysis Force
Salamander
Gradius II

Oh, god. I can't even think of where to start.
Punch Out,
any Ninja Turtles game, especially 2 and onward
If you see a game called Trog! snatch it up, I've been trying to find a copy for a while now. It was a sleeper hit.
Any Mario, Mega Man or Zelda, of course.
Final Fight


Honestly, I would just start grabbing cheap games by the handful and give them a try. There's a lot to discover and the prices are kind of random, a more expensive game isn't always a better one so just buy a lot of cheap ones.

Get Crash n' the Boys Street Challenge. It's got the characters from River City Ransom it it. Get that too. Get Uninvited, Dejavu, Shadowgate, Maniac Mansion, Double Dragon 2, SMB3, Mike Tyson's Punch Out, Solar Jetman, Elevator Action, and Zelda.

Ice Hockey is surprisingly good. For an old game, its actually pretty versatile and takes some skill. Also you can start fights.

ROCKIN' KATS

Pure NES bliss and something of an underground classic.

Xexyz
Mega Man 2
Little Nemo
Crytalis
Castlevania
Gargoyle's Quest 2

Oh, and Bubble Bobble!

I can almost guarantee that any worthwhile games are going to run more than you're willing to pay. They're as much collectors items as anything else now.

Super Mario Bros 3

Crystalis

And definitely Pokemon Gold For NES

Legend of Zelda
SMB 1-3

Super Mario Bros. 3
Mike Tyson's Punch-Out!!
Kirby's Adventure
Star Tropics
Star Tropics II: Zoda's Revenge
Mega Man 2
Mega Man 3
Bionic Commando
Gargoyle's Quest 2
Ducktales
Darkwing Duck
Contra
Super C
Castlevania
Castlevania III: Dracula's Curse
Gradius
Lifeforce
Teenage Mutant Ninja Turtles II: The Arcade Game
Teenage Mutant Ninja Turtles III: The Manhattan Project
Double Dragon
Double Dragon II: The Revenge
Ninja Gaiden
Ninja Gaiden II
Blaster Master
Batman
Crystalis
The Guardian Legend

River City Ransom
Star Tropics
Metal Gear
Contra

Felix the Cat
Super Mario Bros
Contra
Super C
Castlevania and Castlevania III
Duck Tales 2
Bonk's Adventure
Little Samson
Teenage Mutant Ninja Turtles III
Ninja Gaiden and Ninja Gaiden III
Metroid
Legend of Zelda
Mega Man II and Mega Man IV and Mega Man VI
Mighty Final Fight

I'm sure there are more, probably some real obvious ones, but they aren't coming to me at the moment

Teenage Mutant Ninja Turtles 2
Duck_tales2
Kings of the Beach
MEGAMAN2
Rampage
Ninja Gaiden 3 - The Ancient Ship of Doom
Super Mario bros 2 & 3
Battletoads (hardest game ever)
Contra
Donkey Kong

These are the games I played when I was younger. Good times.

Gyruss, Life Force, Zanac & Gun Nac. I'll always remember Zanac as that creepy game that keeps getting smarter until it finally manages to kill you.

Double Dragon
Bubble Bobble
Wrecking Crew
Super Mario Bros (duuuuh)
more