Many people choose to keep working past 65, and keep their coverage under their employer’s group plan. But if you’ve been paying into Medicare via payroll deductions, you may as well enroll in Medicare Part A (hospital insurance) when you’re first eligible, as you’ll pay no premium.
For Part B (medical insurance), most people need to enroll when they turn 65 to avoid a late-enrollment penalty. Only people who have health insurance from their (or their spouse’s) current employer may be able to delay enrolling in Part B – and delay having to pay the monthly Part B premium. In 2020, that premium is $144.60 (or higher depending on your income).