I have got this old Casio fx-92B calculator I am not using any more. I opened it up and found this:
I am not an electrical engineer. Is there a possibility of tweaking this board to make it run custom code?
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No. Just no.
This is not possible in any sense you mean for your level of knowledge about microcontrollers and that specific product.
Even someone skilled with microcontrollers would need specific information about that product to create any useful custom code. This information is not something easily found by merely inspecting the board, and the manufacturer is very unlikely to release it.