My refund is showing that I am claiming the American Opportunity Credit but I did not fill out form 8863. Is turbotax going to let me file without having filled this out?

The TurboTax program generates the 8863 and includes the reporting details you provided through the Education Interview. The form will be included with your tax return. 

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"The American Opportunity Credit can save you up to $2,500 in tax for the education expenses of each eligible student. To qualify, the student must pursue a degree at a school that is eligible to participate in the federal student aid program. The credit is only available to students in their first four years of attendance, who enroll at least half time for one academic period during the tax year and do not possess a felony drug conviction.

The credit amount includes the costs you incur for tuition, fees and course-related books, supplies and equipment necessary to attend the institution. If the credit amount exceeds the amount of tax you owe, you can receive up to $1,000 of the credit as a refund."

More information from TurboTax about Education Credits