I like to use Dunbar's Number (100-250) to approximate the levels of heirarchy in human organizations. The idea is that these organizations are most efficient when organizational layers don't exceed ~150 elements, due to the implementation details of the human brain.
Basically, you can do log_{150}(N) to get a very rough idea of how complex the organization of N people should be. This works for small startups and entire countries. Of course, startups should probably get comfortable with the idea of teams well before hitting >100 employees. Teams can then scale into departments (with new subteams), and once there are many departments, add regional layer, strategic/executive layer, and so on.
One interesting fact is that the USA population has roughly increased by a multiple of Dunbar's number since its organizational structure was codified in its Constitution. Perhaps time for another look?
I have had that exact same idle thought.
In 1813, each of the 182 US Representatives represented on average ~40,000 Americans. Today, each of our 435 Reps stands for about ~760,000 people. That's over an order of magnitude growth. To keep the same rate of representation, we'd need to have over 8,000 Representatives, which is clearly too large a body to get anything done.
So we're probably well beyond the point where we could benefit from a large House of Subrepresentatives and then a smaller House of Superrepresentatives that aggregate them.
Remember, it represents the total number of stable social relationships a person can maintain. If you're looking to allow your employees to have personal lives, you'll want to leave ample room for their family and friends.
Maybe an important question to ask is, how much of your employees' social-emotional carrying capacity is it appropriate to consume? If 10%, then 15-25 is your number. If 20%, then 30-50 is your number.
It's also definitely a upper limit rather than lower limit. Big bureaucracies with many layers of management and small teams can work well, but no one can really individually manage 1000 subordinates.
In practice if a menu has more than 3 levels the user experience will suffer. I keep them at 2 submenus max, possibly 1 if I can get away with it.
Not to mention most menus are not dynamic, meaning they can be just a JSON file or simple HTML.