How to Measure Very Large Capacitance e.g. Super/Ultra Capacitors
From the images of the cell in test setup photo they appear to be similar to the Maxwell DuraBlue line of Ultracapacitors. See this datasheet for more information.

The Maxwell application note 1007239, Test Procedures for Capacitance, ESR, Leakage Current and Self-Discharge Characterizations of Ultracapacitors, may be helpful.

This line of "supercapacitors" has a maximum working voltage of 2.85 VDC and a typical capacitance of 3400 Farads. Most other "supercapacitors" in this type of package have a maximum working voltage of 2.7 VDC.

Be cautious an internal short in these devices can result in a spectacular failure event. You may want to have non-conductive, non-water based fire suppression system available (sand, chemical, CO2, Halon etc.).

Based on the posted test setup photos you will likely melt the alligator clips before you exceed the maximum charge or discharge current.