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Even just a basic multimeter is absolutely essential, you need it to test fuses at the bare minimum. Slow blow fuses especially can look OK-ish while being totally dead. It's also what a beginner needs to check for disconnected wires / broken traces etc. Also it's essential when getting help on the forum for more technical problems - we can frontline help with a lot of issues if we can at least get voltages and continuity tests. So a multimeter isn't just for a DIY-er, it can almost be the passport required for getting online help and diagnosis without (or at the very least, before) getting it to a professional.
For me #1 mobile phone camera for the many pics needed to reassemble what has been disassembled, #2 (cheap) soldering iron for coil wires etc, #3 DMM, #4 Dremel