Is there a limit to how much a retiree can earn?

Theres no limit.  Are you asking about Social Security?   There are 2 different things to know about social security. People get them mixed up all the time.

1. Your actual SS checks

If you are over full retirement age your actual ss checks won't be reduced. Otherwise they will actually reduce your payments if you make too much other income in the prior year.  See SS FAQ for working after retirement

https://www.ssa.gov/planners/retire/whileworking.html

And see the special first year rule

KA-01927

And Benefits Planner: Retirement | Special Earnings Limit Rule | SSA

2. Income Tax

For any age up to 85% of Social Security becomes taxable when ALL your other income plus 1/2 your social security reaches:

Married Filing Jointly: $32,000

Single or head of household: $25,000

Married Filing Separately: 0

Great question!