I am currently using an arduino coupled to a mosfet to drive a 3v DC motor. What it does, is turn on the motor for a second, wait 10min, turn it on again for a second, and repeat the next day. I would like to go a step further, and replace the arduino by a simpler setup, but I don't know what circuit I'm looking for. I feel like it should be a modification of something that generates square waves, just with a much longer period.
If you want a non-micro solution you might want to try MAXIM INTEGRATED PRODUCTS ICM7240IPE+ IC, TIMER. You will need to add an extra monostable (555) chain to provide your desired waveform. Be very careful when using the 555 to generate long delays - Electrolytics have wild tolerances, high leakage currents and poor temp stability. If the leakage of your timing cap exceeds the charging current, the device will never trigger!
Alternatively you could code it into some small PIC device with an internal oscillator, or even load the compiled arduino code into an AVR micro.
The PIC solution is the cheapest by far, but you would need to code it in C or assembler.
The AVR solution would enable you to use your existing working code, but you would need to provide any necessary supporting components (oscillators, voltage regs, etc)