How can I find how many years I received the American Opportunity or Hope Credit?

It won't tell you on Form 8863 how many times it was used.   You have to verify that yourself; you may need to check past tax returns to be sure.    Form 8863 only calculates the credit, but it calculates it based off your answers to the questions.   If the student has taken the credit 4 times, they are not eligible to take it again.   There are other qualifiers as well - See Am I AOTC Eligible?    If you answer that you have claimed it 4 times already, you won't qualify.   

An eligible student is defined as a student who:

• Is enrolled at least half-time in a program leading to a degree, certificate, or other recognized credential
• Had at least one academic period beginning during the year
• Didn't claim the American Opportunity Tax Credit for more than three previous years - 4 is the max
• Didn't complete the first four years of post-secondary education before the beginning of the year