The master gardener recommendation is around 3" for cooler season grasses and around 1" for hot season grasses like bermuda and zoysia.
I cut my grass at 2 1/2-3" during the summer and 2" at start and end of season.
I cut the trails, which are mostly wild fescue, down along the creek at 2 1/2". Stuff grows great, no watering, no fert, nothing.
I throw on milorganite in the fall onto grass here around the house, approx. 1/2 acre of fine green mixture mess but looks good. The green includes some dandelion, chick weed, crab grass, lots of good green stuff.
Ralph