I’ve seen a cat jump up onto the top of a shed roof that was something like 6"8 or so above ground, and where the roof overhung the front wall of the shed, so there was no vertical surface for the cat to brace or scramble up - basically it jumped up onto a ledge.
It’s possible that the cat actually got its front claws hooked onto the edge as it jumped and sort of leapfrog-boosted the last 12 to 18 inches or so, but it did so in a way that made the entire jump look like an effortless single leap.
I don’t have the event on video, and I think it’s probably pretty unusual - a cat isn’t going to choose the hardest jump in general - it’s going to choose an easier one, if that’s available.
Yes, for example, why really push it to get to the top of that door in a single bound when you can hop onto a table, then an unsuspecting passer-by, and from there to the door?
If you want video evidence of how high a cat can reach or raise her centre of gravity jumping straight up, those Oscar the Cat videos should be pretty convincing since the guy measured the distance.
Unless you undertake a significant DIY project, you’re not going to be able to keep all four cats contained. If a cat is determined enough, they’ll find a way. A lot will depend on the temperament of the cat. Are these outdoor cats normally? Expect them to try everything they can to get out. Are they indoor cats? They may be satisfied if they have someplace they can just sit and watch what’s going on outside.
One thing is certain, you may be able keep the cats in, but you can’t stop the squirrels!
Like standing next to the door/gate and saying, “Meow. Meow. Meow. Meow…”
One day my father was letting dogs outside (into a fenced yard, but one fenced to hold dogs in, not cats) when he happened to notice that one of the dogs had too many legs.
One of the cats was walking underneath the dog, hoping to get through the door and outside. Almost made it, too.
I recently saw an ad for a device that is basically a roller bar that you attach to the top of a fence to keep the cats from being able to grab the top to finish pulling themselves up and over.
A little pricey IMO, but looks like it would work if it’s in your budget.
My indoor cat who’s desperate to be an outdoor cat does the exact same thing. My dog sleeps at night in the garage where it’s warm and roomy (I don’t keep my car in the garage) but every morning there’s the transition from garage to indoors to backyard. My cat will sometimes walk underneath the dog during the transition hoping to get out too, but usually it’s him hiding underneath a table close by only to try to dart outside the moment the I open the door for the dog to leave.
My house cats have their own cat yard, homemade and cribbed off of Purrfect fence.
I used 8 ft T-posts, a sturdy plastic mesh that’s the same sort Purrfect fence sells, big shelf brackets for the overhang, and snake-type pipe clamps to hold those to the posts. I also used a gazillion zip ties, and some flexible wire in an L on the ground to keep my rotten terrier from digging in.
Here’s a picture of the 1st one I made (it’s a public FB post, hope it works)
And this is a short video of my current yard. It’s hard to see but the mesh does have an overhang.
I’ve only had one very feral cat get itself in to the old one, because there was a porch railing nearby and he walked on the overhang and dropped in. He is also the only cat to get himself back out. The mesh is just floppy enough that they don’t really want to mess with it. The holes in the mesh are also big enough that birds and small squirrels don’t get trapped.
Since I have 11 cats in a 1500 sq ft house, this has made everyone a lot happier!
My cat used to be able to clear jump 5’ vertically and the same distance horizontally without a blink; it was quite startling to watch that moment in flight when she would just arc through the air, legs dangling. She’s slowed down a little in her old age but still jumps pretty well and often scrambles straight up the 7’ wooden fence between our yard and the neighbors’ yard with ease.
I think six feet is high enough. 
Hah! The little guy couldn’t get a grip on the floor. They should have filmed him trying this day after day as he grew. Maybe they did but they mysteriously disappeared after unexplainedly having a massive supply of cat food in easy open pouches delivered to their house.
We have a small yard, and most of the fence is only 5 feet tall. I put up bamboo fencing along the entire fence, then attached plastic roofing panels on the inside. It’s ugly, but it does the job. The bamboo is to cut down on the ugliness from the outside.
You can see some of the setup out the window in the picture. That’s Irving, Rocky and Opehlia in the foreground.
http://www.mackeypdx.com/publicpics/catsnfence.jpg
(The picture wouldn’t show up inline when I pasted the link on its own line, I was just getting a little error icon, WTH?)
Wow, a real life crazy cat lady! Nice to finally meet one 
.
I lost an elderly girl last year so I did have 12 in the house, plus there are four more barn cats that came with the property.
I have 15 cats and would have more if I could do it and keep everyone happy. CCL all the way! 
The problem with using kitty videos to determine what cats can and cannot do is exactly that the cats in the videos don’t have to jump as high as they possibly can; they are not in a desperate escape situation that calls for desperate, all-out action.
The squirrels in my neck of the woods make cool multi-meter jumps from tree to tree when the distance between trees is long but not too long. When the distance is too great, they go down the tree and proceed from there. You never see the spectacular, desperate, barely-making-it 5-meter leaps from tree to tree, except when the squirrels are running for their lives, from a predation attempt.
Yeah, I mean, it’s sometimes possible to tempt a cat to jump high by offering it a treat it really likes. My sister had a cat that loved whitebait and could jump high enough to hook a claw into them even if I held them as high as I could reach - I’m 5’11" - so the cat was jumping to reach about 7’ or so - obviously, that’s not the cat raising its entire centre of mass to that height - just the tip of one claw, but also, I’m not sure that was the maximum jump for that cat anyway.
I notice nobody has mentioned concertina wire or razor wire.
Probably because it’s not remotely helpful or funny.
It’s outright dangerous, cats can use that stuff on you if they want.
I see you’ve met my cat.