There certainly is a charge cycle counter!
% ectool battery
Battery info:
OEM name: NVT
Model number: Framewo
Chemistry : LION
Serial number: 00CD
Design capacity: 3572 mAh
Last full charge: 3399 mAh
Design output voltage 15400 mV
Cycle count 89 <========
Present voltage 17526 mV
Present current 3 mA
Remaining capacity 3399 mAh
It’s not currently surfaced through ACPI, however, and I’ve requested that it do so: Feature Request: ACPI `_BIX` to report battery cycle count