The mid-level exception (MLE) is a yearly allowance that lets a basketball team over the salary cap sign a free agent up to a set amount. Teams in the luxury tax get a smaller taxpayer version.
In HoopSim the MLE is $14.1M by default for up to 4 years, for teams below the tax line. At or above the tax line it becomes the taxpayer MLE, $5.69M for up to 4 years. There is also a low-level exception (LLE): $5.13M for up to 2 years, for any team below the second apron. Each works once per team per league year, from the opening of free agency through the following season. You pick the exception on the offer when you bid, and it comes back if the player signs elsewhere. No exception signing can pass the hard cap. HoopSim has no room or bi-annual exception.
Rulebook: Exceptions. Related: soft cap, luxury tax.