Earlier quoted context omitted.
You can easily store notes on a TI calculator. I had a TI-85 in college, and was occasionally guilty of writing TI-BASIC programs to make tests easier (for example, I had one which computed Euler's method).
Most test proctors make you clear your memory before a test in high school. In college they don't care as much because they are used to making "open book tests".
That said, I don't recall my teachers ever getting too fussy about having us clear our memory; in fact, iirc one of them actually told us to write programs to handle some of the nastier formulas.
Anyway, in my opinion, a good mathematics course should focus less on regurgitation of formulas, and more on problems solving---recognizing which theorems or strategies might apply to a problem and applying them to come up with a result.