1. Beat em ups are usually relaxing because they're a forgiving genre, you have a lot of room for mistake so you can just shut your brain off and enjoy the games.
So games like
- Turtles in Time
- TMNT Hyperstone Heist
- Final Fight 1, 2 and 3
- Streets of Rage 1, 2 and 3
- Ghost Chaser Densei
- Warriors of Fate
2. Now this might come as as surprise but Batsugun and many post Batusgun shmups
If you don't play them for a high score they can be VERY relaxing just crank up the number of lives and special attacks, set the difficulty to the easiest possible and you're good to go. Just beautiful music and fun relaxing gameplay.
- Psyvariar 2 - PS2 version, if you can't run the PS2 version on real hardware play the DC one then, emulation isn't good for PS2 version but it is the most relaxing out of all versions
- Trizeal
- Batusgun
- Dangun Feveron - this is also on PS4 now
- Super Aleste - SNES
- G-Darius
- ESPGaluda 1
- Mushihimesama
3. I personally find Sonic 1, 2 and 3 to be exceptionally relaxing as dying is pretty much only possible in the last few levels, they barely have any bottomless pits, they have frequent checkpoints and smooth, relaxing gameplay.
Sonic & Knuckles starts out relaxing but then comes Flying Battery zone and it's anything but sadly.
You might also want to check out Sonic 3D or 3D Blast a it's known in the US, that can be boring but at the same time very relaxing.
4. Atari 2600 games
These games are short, very repetitive usually and VERY easy to understand, pick up and play for 5 minutes then move on. The absolute perfect games for pick up and play gaming on the go or for quick relaxation.