Linux distro with just busybox and bash?

Ttylinux

What you are searching for is called ttylinux.

Major components

• Kernel
• Glibc
• Iptables
• Dropbear (scp, ssh, sshd)
• GPM
• Bash
• Busybox
• E2fsprogs

Not maintained

The distribution was hosted at Ttylinux.net, and is currently not maintained.

Download

Latest ttylinux release is dated 2015.02 (updated 2015.03).

A good linux distro that I've fallen in love with (and also has no GUI) is MicroCore Linux. It also has an older brother TinyCore Linux that does have a GUI (a pretty crappy one though).

You could always try DSL (Damn Small Linux)

I don't think any of these explicitly come with busybox (I don't know of any distro that ships with it), but it shouldn't be too hard to install it afterwards. From what I understand your main concern is a small fast system with a decent shell.

"Alpine Linux is a security-oriented, lightweight Linux distribution based on musl libc and busybox."