Portable Linux Shell (no admin)

So, since my high school lets me use whatever I want to program things in computer science class, I've decided I want to put bash/cygwin-like environment on a usb stick. The only problem is I dont know what would be the best way to this. It definitely isnt possible Win10 bash and I dont know how portable cygwin is. There are 2 issues I'll need to get around: 1. Windows fucking up the driver letter 2. Not being an admin at school. I need to be able to run this without admin privileges

Git for Windows comes with a Bash (MingW instead of Cygwin though) and is available as portable version.
https://git-scm.com/download/win

I think MobaXterm is portable and cygwin based, but my favourite for getting nice colours is ConEmu and is native Windows.