It seems like we ought to only purchase free time when it comes cheaper than a certain figure, $x/hr, and ought to only work if we can sell our free time for more than $x/hr.

Time is not fungible (it's not like most people could decide to work 80 hours one week and 0 hours the following week -- they'd get burned out the first week and bored the second week; also, it's not like it's easy to find people to socialize with at 5 a.m.), so the value of x would depend on the time of the day, day of the week, how much stamina you have, and how bored you are, among other things. In certain cases it can even become negative (i.e. people willing to pay money for anything to keep them entertained while they're waiting for something and have nothing else to do).

Anyway...

An example that springs to mind is whether to drive, walk, cycle, or use public transportation (though how much money and time you need for each depends on where you are, you're trading off safety as well as time and money, and being able to comfortably study and/or sleep on buses/trains (not everybody can do that) can reduce the effective time cost of public transportation for long routes). Another one is whether to rent an expensive apartment near your workplace or a cheap one far away from it.