I only keep a couple $K laying around. From most of the posts on this thread, that seems about right.
Last year, one of my horses incurred about $4,500 in vet bills in about six weeks. That's the biggest emergency I can think of. But the biggest chunk of that was surgery, which I had a couple weeks to prepare for, so I actually used it for a bit of credit card churning. In the case of an immediate veterinary emergency, the equine hospital requires payment "when rendered" but really allows 30 days from billing. This amount actually exceeds the EF I had in place at the time, but it wasn't too difficult to hoard some cash to get it payed off by the time the CC payment came due.
The paradox of EFs is that, in general, the easier it is to build one, the less you really need it . As a Mustachian with a 50% savings rate (and assuming job stability), most small emergencies would just mean saving less in a given month. I don't want to go in and monkey with my pre-tax deductions to cover a $1,000 unexpected expense, but would do so to refill my EF after a rare $5K expense that zero-ed out my EF and my savings rate for a month. If you're already spending nearly everything you make, those $1K unexpected car repairs or plane tickets or whatever create a much bigger speed bump.
Another piece of the puzzle for me is that I'm a mid-career fed, so job stability is great and I have a shitload of sick leave saved up. I've also got a few assets that would be sold off in the case of a severe emergency like a major medical problem (e.g. if I get cancer, I'd need to sell my horse, truck and trailer, which would liquidate around $25K or more). DH is also a mid-career fed, and he likes to keep more liquid cash around, so if our septic fails or our roof caves in, he could just write a check for it.