Probably CR2032, perhaps the most common coin cell battery. Like any other dead battery, annoying but you can get them at the dollar store. My digital caliper eats batts like OnceADay vitamins (no auto-off feature, I think) and a handful from the dollar store has saved the day before.
Yeah, I got the last two digits transposed. It’s 2032.
The wires are very thin. They don’t interfere with the door being closed. The base unit is outside the oven and can be read from your counter. Here are the basic types of digital thermometers:
On the left is a handheld thermometer. It’s used for an instant read and is not left in the food while it’s cooking. Next is a basic probe thermometer. It has a wired probe that goes to a base unit. The probe stays in the food while it is cooking. The base unit shows the temp and you can typically set an alarm at a certain temp. Next is a wired probe with a wireless remote. The probe is wired to the base unit which just shows temp. It communicates with a wireless unit you can take with you and it will have alarm functions. Last is a wireless digital probe. It communicates with a base unit and your phone. The base unit will typically just show temp. For alarm functions, you’ll need to use the phone app.
For the most reliable use, I would recommend the kinds of wired probes in the 2nd and 3rd pictures. Being wired and using regular AA batteries, there are fewer issues to worry about with regards to range or the unit being dead. If you grill, get a wired probe that is rated for grill temperatures. Probes rated for grilling will have a metal shielded wire. A grill probe will work fine for both oven and grill.
I’ll stick with my analog and wireless digital mix. (Oh, and my beloved thermo-pen, but i don’t use it as a probe.) It works for me. Fwiw, I’ve never had a thermometer run out of power mid-cook. And since upgrading from the original meater block to the combustion, i haven’t had any issues with dropped signals, either. (And i got an audible warning when i lost the signal, so it wasn’t a huge deal.) I’m actually extremely happy with the combustion probe. It’s slim, it has a bunch of sensors so you don’t have to worry about inserting it “just right” , it gives mildly useful info about the temp of the oven, and on the surface of the roast, and it has a killer “has this been cooked to fda food safety standards” app that allows me to confidently serve juicy chicken that hasn’t been over-cooked. When my last one died, i really really missed that app.