How many suspensions can you get before you're banned from Xbox?

Solution 1:

Microsoft's official policy is:

We may permanently suspend your profile if we can no longer trust it due to a severe violation or if our attempts to correct repeated negative behaviors are unsuccessful. A permanent account suspension blocks your profile from accessing Xbox Live in any way.

Types of Xbox enforcement actions

Only Microsoft can answer specifically what that means, and odds are its proprietary anyway. The answer is almost definitely "it depends". Three suspensions in the first three weeks of a new account is very different from three suspensions, years apart, on a 10 year old account.