whats the simplest roll-your-own solution?
For charging only, the battery's phone. For charging and logging, a Rigol DP832 Power Source with a small SCPI script for logging, and an improvised connector for the battery.
For discharging, two 15ohm/8W resistors in parallel and a Metrix mtx3293 DMM that can log both the discharging current and the battery voltage simultaneous. For very old batteries, a thick old pot was used against possible fire/explosion damage.
As a fun tale, one Nokia battery that was tested was from a phone lost on a very circulated road about 10 years ago, after many cars drove over the phone. There are dents and bending signs all over the battery. The phone was damaged even worst. That battery was forgotten until a few weeks ago, when it was measured. The battery was still about 1/4 charged after all that time. With all the dents and its very old age, that battery is still working at about 70% of its original capacity.