Just about the simplest circuit using the AD620 that I can think of, that should amplify a positive waveform:

Simulation looks good, a perfect x2 amplification:

But on the breadboard I notice a strange thing: when Vs- is higher (less negative) than -3 V the top of the output waveform levels out. This is even the case when I shift the input signal upwards and increase Vs+.

The datasheet shows ├▒2.3 V at 'Operating Range'. I suppose that means Vs- should be no higher than -2.3 V for the AD620 to perform normally.

This is with the AD620 supply pins set to -4 V and +8 V. Waveform looks normal:

This is with the AD620 supply pins set to -4 V and +7 V. Clipping at the top as expected, since the chip does not go to rail.

This is with the AD620 supply pins set to -2.5 V and +8 V. While there is plenty of room between the expected maximum and the rail, the waveform gets more shallow and 'ugly'.