Is there any way to automatically replace a formula with its value
when a condition is met, without the user changing it manually? I'm
not talking about macros, but rather, for instance, if NOW()=A2 then
the value in A3 stops changing.

Apparently, there's a formula in A3 but it's value is supposed to freeze at
a particular time.

One thing that might help is to first use
Tools Options Calculation
and check "Iteration" and set "Maximum Iterations" to 1. This allows
circular references.

Then, in A3, put
=IF(NOW()A2, A3, your_formula_here )