BEST Retro Gaming Handhelds for Your Buck!
Tier 1 ($60-$100) - Budget Handhelds
Anbernic RG35XX Plus
Screen
3.5-inch IPS full viewing angle, OCA full lamination/ 640*480
CPU
H700 quad-core ARM Cortex-A53, 1.5GHz frequency
GPU
Dual- core G31 MP2
RAM
LPDDR4 1GB
3300mAh Battery (Up to 8hrs)
Anbernic RG35XX H
Hardware is the same as the RG35XX Plus, now in a landscape form with added thumb sticks and design tweaks! They also offer a metal design for a few more coins if you're into that sort of thing. I think this is an excellent entry handheld worth a hard consideration at it's price-point! The dual thumbsticks make it more enjoyable to play systems such as N64, Dreamcast, PS1, and some PSP!
TrimUi Smart Pro

The TrimUI features a 16x9 4.96" 1280x720 display making it the largest screen in my recommendations under $100! Some settings may need to be adjusted to play PSP as it's not quite powerful enough to handle all games at default settings. Everything prior should be capable of playing (if you don't mind black sidebars or widescreen stretching!). It's also worth noting that it lacks L3+R3 functions and it's not a touchscreen, so keep that in mind when selecting games!
HONORABLE MENTIONS
Most Pocketable - RG Nano Pocket Mini
Powkiddy RGB30
The Powkiddy RGB30 features a 4" 1:1 aspect ratio 720x720 display. I believe this runs on Linux OS (not Android like previous models above) which some people prefer. I've heard it runs up to PS1 just fine, and they got a lot right about it's design (such as it's start and select button locations!) It give's you lots of screen real estate for older games while maintaining a smaller pocketable form factor!