As @Kenneth Garrett mentioned previously, the "standard" advice is to hold back 6 months worth of operating expenses in reserve.
I would add that most of us, as we acquire more doors, feel that 6 months is too much. It's the "acquire more doors" part that is key here. As you get more doors you begin to notice that "problem$$$" usually only occur on one or two properties at a time. And that very often the income from the other properties covers the cost of the issues on the few problematic ones.
I deal with a a fair number of property managers, and they will generally estimate 3-5% of rent as being needed for maintenance. This number is WAY to low (for most of us). The way these PMs come up with that number is by cost averaging maintenance costs over the hundreds (sometimes thousands) of properties they manage. I've tried to explain why standard deviation is an important consideration in this calculation but, as you might imagine, it ends up being too mathy for most PMs and they tune out. :D The problem is that of their hundreds of homes they manage, the average cost is 3-5%, but some poor owner is stuck buying a new AC each month.
But the cost averaging principal still applies. As you get more homes, you will find that you don't need to hold back quite as much.
So how much do you hold back? Ultimately it's a personal decision. It does seem to me that there is an upper limit. Questions like: What is my insurance deductable? What is the longest period my PM has needed to rent a unit? What is the largest repair expense my PM has seen? Those types of questions would likely help you find a reasonable upper limit on how much you should reserve.
I would love to a large PM share their data with a mathematician, and probably a professional poker player :), and see if they could come up with some guidelines.
