Like Black Tiger said, there are many threads in this forum dedicated to this question, but I'll throw out a few that I like which may suit you.
Action type platformers
Bazaru De Gozaru - not a platformer, but this puzzle game's cute monkey does jump from platform to platform in many stages. O:)
Doraemon - Nobita's Dorabian Night - had both a HuCard and a CD release
Fausette Amour
Gotzendeiner - not a platformer. It's an action/adventure.
Horror Story - a forced scrolling run-'n-gun platformer
Iga Minden Gao
Mystic Formula - not a platformer. It's an overhead run-'n-gun.
Renny Blaster - not really worth it's expensive price tag.
but it is worth a playthrough
Shoot-'em-ups
Avenger
Cho Aniki - if your mind can handle
the weird graphics and music, then is is a fine game
God Panic
Hellfire S
L-Dis
Macros 2036
Nexzr - not easy per se, but it has unlimited continues :-"
Sapphire - a must-play shoot-'em-up, although it requires the Arcade Card
Spriggan Mark 2
Star Parodier
Steam Heart's - has hentai cut-scenes which can be skipped, but the game itself is actually good.
Super Raiden
Sylphia
Zero Wing - not easy, and
may be unbeatable for those lacking skills (like me), but still a great game for the 8 (out of 10) levels that I can play