OP has not told us anything other than just how valuable his time is.

Knowing the carrier and departure airport would help.

Most carriers, including DL, UA, and AA if this is US domestic, all specify that one must be at the gate within X minutes of scheduled departure. Thus, unless you are at some small outstation where there is no realistic chance of an aircraft substitution, you put your ticket at risk if you no show and have not checked with your carrier.

In particular, estimates for aircraft repair, weather, and ATC delays are all subject to change. Even at a small outstation, there may be a ground hold for the inbound flight. But, that could disappear and your 3-hour delay turn into 30 minutes in a split second.

So, you have to decide whether your time is too valuable to waste waiting for a delay or waiting for the next flight.