The last two generations of Samsung NX cameras were built around Tizen Linux, and it was (and still is) easy to get a root shell on them. They still make great photos and you still can buy them used for a good price.

NX300/NX30/NX2000 had a read-only rootfs, but for NX500 and NX1 there was a persistent mod that extended camera functionality with a menu, and you can actually SSH into them and rsync your photos... while shooting!

Background: I've recently taken over maintenance of the NX-KS mod at https://github.com/ge0rg/nx-ks-mod