Like absolute zero, is there an absolute maximum temperature?
Possibly... there is something called the Planck temperature where it is speculated current theory may break down, that being said, what is your motivation for asking if there is a maximum temperature? For any theory speculating a max temp is pretty different than the reason for absolute zero.
I'm wondering if absolute cold (or hot) is where 'time' no longer makes sense?
It depends on how you define your system and what you allow to be called a thermal equilibrium. In statistical mechanics, temperature is defined as dE/dS. In normal systems, the entropy increases as you increase the energy because you have more quanta of energy which you can assign to your degrees of freedom. However, in a restricted system with just a limited number of degrees of freedom and a maximum energy state, the degrees of freedom start to fill up as you increase the energy past halfway, and the entropy decreases with increasing energy, which indicates a negative temperature. In this restricted system, negative temperatures are hotter than positive temperatures, and the limit of hottest temperature is -0 (zero, approaching from negative side).
So, the scale of hotness goes (from left to right)
0 ... 1 ... +infinity = -infinity ... -1 ... -0
An example of a restricted system is something like a spin system, where you have a bunch of particles with spin up or spin down, in a magnetic field, so spin up has higher energy than spin down. The highest energy in this system is if all particles are spin up. So, the ground state, with all particles spin down is near absolute 0, and all spin up is near absolute hot (absolute negative 0?).
A few caveats are in order. The temperature of the restricted system applies to just this restricted system, whereas we are not able to assign a temperature to the "full" system because the full system is not in thermal equilibrium. The energy stored in the spins will eventually spread into other degrees of freedom like motion, and the total system always has a positive temperature, as far as I know. It only makes sense to define a "spin temperature" if the spins interact with each other much more strongly than with external degrees of freedom (e.g., spin energy leaks into motion energy very slowly).
In fact, if you want to talk about true thermal equilibrium, then you need to include pair production and annhilation degrees of freedom, and wait until the universe settles into heat death. So lab experiment is far from true thermal equilibrium and is using a restricted temperature measurment.
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