Embedded system-current draw logging
The $30 Tekpower multimeter has an RS-232 interface and would do that with a laptop and the appropriate software. Connect it in series with your +5V (or 3.3V or whatever it's running on) and come back in two weeks.

Note that the meter does not send ASCII strings (which most any terminal program could save to a text file) but rather a packet of bytes representing the display, segment by segment. I use QtDMM but it may not have enough storage for two weeks worth of data. Some terminal programs may be able to collect and store binary data which you could post-process.