The 8bitdo Pro 2 came with both a rechargeable battery pack and a battery holder for replaceable batteries. Works seamlessly with Linux and Steam via D-input. No personal experience with the Pro 3 to say how it differs.

Yeah I just use an Xbox one controller with rechargeable batteries and it's flawless.

An Xbox controller. Probably not that hard to replace the batteries in 8bitdo controllers. 4 screws on the Ultimate 2C. I'm waiting for the new Steam Controller though pictures of it look like it may not have AA battery slots like the first one

I use a Xbox controller and works perfectly with Linux.

Xbox

I have an Xbox 360 controller that has been awesome, takes AA batteries.

my original Xbox One controller has been seamless with my Linux Mint setup, and it runs off of two AAs

I'm very happy with the "Flydigi Vader 4 Pro" I got off AliExpress. It can connect through Bluetooth, a plug and play dongle, and USBC.

my Xbox elite series 2 works a treat, just has to stay plugged in unless you wanna spend like 30$ on a dongle sold separately

Maybe a knock off wii pro controller? I dont think there really has ever been much demand for such a combination, it was simply cheaper to not include rechargable batteries 15 years ago.

Wii Remote, hands down. First of all, most Linux distros have out-of-the-box support for it. Second, you can connect a bunch of accessories to it, even lethal-sounding ones, such as the Nunchuk! Third, it's perfect for mouse emulation, and all you need for it to work is to put a couple of candles under your monitor. Nothing beats the Wii Remote. There's the small problem of most computer games not supporting the Wii Remote out of the box, but that's a minor issue, really.

EDIT: Almost forgot. Let's say you need a controller with a conventional layout because of a pesky game that can't be played without it. Just slap a Wii Classic or a Wii Pro controller into the Wii Remote, and bam, you got exactly what you needed.

(I'm sorry if it's a bad joke.)

I hope the Steam Frame controllers will be sold separately. They have replaceable batteries and you're not forced to hold your hands in one position.