How many times can you charge your phone without damaging the battery?

Charge it as often as you like whenever you like. You won't damage the battery.

KiltedTim's advice is absolutely correct. Charging a battery doesn't damage it. The battery will, over time, lose capacity. That is expected and not considered damage.

You may also want to read this tip by another forum user:

When to charge your iPhone or iPad - Apple Community

Depends on what you mean by "damaging" the battery. Rechargeable batteries inherently degrade during charge-recharge, and there's really nothing that can be done to stop it. It's never put back exactly the same way it was before. There's always a tiny bit of outgassing and it never re-forms perfectly. Probably the biggest enemy of battery longevity is heat.

There are a lot of little things that can improve ultimate battery longevity, but for most users they're impractical. You'll occasionally hear about the 20-80% rule (never let the battery go below 20% or above 80% or variations on it), but for most users it's almost impossible to manage that sort of use without an automated way of programming those limits into battery management.

I mentioned the "20-80% rule". Apple has publicly announced that iOS 13 will contain a variation on this.

iOS 13 Preview - Features - Apple

Battery

Optimized battery charging

A new option helps slow the rate of battery aging by reducing the time your iPhone spends fully charged. iPhone uses on-device machine learning to understand your daily charging routine so it can wait to finish charging past 80 percent until you need to use it.

Not charging up to full capacity is nothing new. I've helped set up a laptop computer where one battery management option was for "maximum longevity" where the charge range was intentionally limited to avoid the extremes of charge. Several electric cars can be programmed to limit the maximum charge, which some owners believe will at least quadruple the longevity of the battery.