How do I create a dump of a Game Boy game so that I can have it for backup as a ROM?
I know that I can download the ROM (though slightly illegal) if I really wanted to, but I want to know "how" ROM dumping it's done.
There are cart readers available for virtually any type of cartridge, including Game Boy, that will allow you to hook it into your computer to be read. Modern ones, like the Gen3 Reader/Writer (supports GB/GBC/GBA) use USB to connect with a PC and also allow you to flash your own blank carts for homebrew software. Alternately, you could also rig one up yourself with an Arduino and a spare cartridge slot, as described here, if you prefer something more DIY.
My experience is from the "old days" of the 80s and 90s, so I'm not sure what is available today. At work we had several PROM burners that could write (or read) PROMs, EPROMs, and EEPROMs. It could likely read ROMs as well, because IIRC that era's ROMs and PROMs had the same pinout. With that equipment, you would need a free device (not installed in anything). You could not attach probes to an in-circuit component to read it. So you can search for this sort of equipment. It would probably be a lot more expensive than just buying another TI-84, though.
A second option would be to find a programmable computer that had sockets for the type of device you wish to read. Then, use a memory dump program to write a memory image of that device to disk.