New roof, new AC/heating unit, new septic tank, complete casualty from natural disaster, can't find a renter for 2 months, etc. A lot of these things depending upon where you live will run around $3,500 depending upon the size and circumstances involving the emergency. My aunt (lives in a completely different region) needed a new main plumbing line going to the city. The cheapest she could find was $7,000. I had the same thing done at mine for half that and the length was twice as long. Go figure!
As you acquire more rental properties and they generate stronger cash flow. You can then start to spread the risk around a little bit more. It would be great to have an emergency fund of equal amount for each property, but you also don't want to sit on too much cash that earns you nothing after inflation. The odds that an emergency will happen to each of them at the same time is less than if you have just one.
CAUTION: if you live in an area where the geography could devastate many of your properties at once (earthquake, hurricane, tornado, flooding, etc.) then this is very possible and actually very likely.
Get estimates to see what it would cost to replace major items and also to see what it would cost to pay the deductible in the event of a complete or partial loss. The maximum amount is what you should keep on hand for this emergency.