What Display manager (DM) are you running?
Step one, hard reboot (hold the power button, or pull the plug if that doesn't work).
when it turns on, see if your DM/login screen launches, if it does just select budgie. If you can't select which DE to launch, get into tty ((ctrl+alt+F2-7, whichever) - skip the next step. If you can't get into your DM, using grub (not sure how, google it, but for refind its F2) you should be able to get to a page with the kernel parameters, add a 3 to the end of that line, that should dump you at a terminal.
At that point, find the options/config file for your DM and find out how to give options for choosing DE, disable auto-login etc. the arch wiki will help you.
If you are running lightdm, and you didn't configure it yourself, you may want to swap it for lxdm, this is one of the areas where manjaro's auto-configure sometimes screws up and breaks things. LXDM is also a lot easier to setup - read the manjaro wiki for a beginners guide on how to do this, then read the arch wiki page for how to configure it.
I'd probably remove i3 at this point, you can reinstall it later, but we don't want to accidentally try and launch it.
Reboot, it should work now, if not report back with exactly what you did.
if you don't understand how to do any of this, read the arch wiki page on pacman, then ask for clarification if you need it.