When I'm at home (as opposed to college), I ride my mare 6 days a week and school over fences once. We don't jump the height she shows at unless we have a show coming up or something along those lines. Occasionally we'll let her do a big vertical or oxer because she really loves her job and gets a little unhappy if she doesn't get to play over the big jumps now and then
I ride 3 days a week, usually hack out on trails at least one of those days (more often in summer when there's more daylight)... when we ride in the ring, sometimes I'll also go over a little vertical during our canter, just to mix it up and keep him paying attention. I'll usually pop over a few (3-5) fences before we finish, but if we do something complex, like a tricky combination or a funny angled line, we'll quit after 1 if he did it nicely... no sense drilling it if he gets it.
I only do local schooling shows and the occasional starter trial; the highest we jump in shows/ST's is 2'6". When we jump at home, we rarely jump over 18"-2' unless we have a show coming up that weekend. So I really don't think jumping a handful of 18" fences 2 or 3 times a week is over-stressing him.
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Well, you cant live by 'a horse has only so many jumps' in him. If you jump a horse only once a week 12-18 inches and then show 2'6" - you are likly to strain something. That is from my experience.
However, I once trained at a jumper barn and a lot of those horses jumped every day and I did not like that.
The best program that seems to work the best - is - my riders jump 2x a week. We do a lot of small gymnastics and turns etc. BUT we finish the lesson over fences at least the height we show at. Basically, if my horses here jump 3 foot courses at home in schooling - they show 2'6". If and when they actually go up in height - that may change. Like a 4 foot horse can school and show 4 foot - I am not going to drive him up to 4'6" to school..... but in general - its more the level of the rider and confidence in the horse. If the horse and rider are confident over 2'9"-3 foot at home - 2'6" is about equivilent when you add flowers and flashy bits to the jumps and add stress....
Also, I have 4 of the 8 horses who do not school well over 12-18 inches. They are too athletic and big. They need 2' at least to pay attention unless we are doing gymnastics or like activities.
BUT to answer your question - 2x a week is pretty good - 3x perhaps if you event or do jumpers and have a show coming up..... of course, jumping is flat work with obstacles in it - so you should spend more time on flat work and making sure your horse has a good attitude (like going on trail rides and hacks too). But I think their legs and bodies need 2x a week to keep their 'jumping legs'. Again, the riding session is 50-75% flat work even when we jump.
Also, I generally like her to be comfortable schooling 3-6 inches higher at home than I'll be asking her to jump at a show. Not meaning that every fence we school at home is that height, of course, but some will be.
My A/O hunter schools 2'9 ish once a week and he'll school at height once or twice before a show. He's a good boy and knows his job, he doesn't need to be working over big fences all that much. He's the type of guy who could have six months of only flat work, pop one 3'6" jump and walk into the show ring. Plus in the winter we're in a pretty small indoor and schooling courses at 3'6" with the lines built to stride can be kind of scary.
My A/A jumper jumps twice a week, but we keep it at/under 3'. With jumpers I think the most important thing to school is the turns and rythm, which you can easily do over little stuff. We'll do the height two or three times in the week before, but mostly our schooling is over small stuff.
Once they're past the green stage, I don't like to drill over big fences. They work hard enough for me at the shows - they don't need to be pushed at home. I'm a firm believer that there is plenty to practice at home over smaller stuff that unless your goal is to build experience/miles/confidence over the big jumps, you don't need to be doing your division's height.
I ride anywhere from 3-7 days a week! I have my lesson and two days dedicated to pony schooling, and the other days I ride whatever my trainer needs hacked or schooled. I always school o/f at least 2 days a week, and that changes if my trainer wants me to school a trial horse or naughty horse o/f for her, but a trainer always has to be with you while schooling o/f at my barn. And it depends on the horse as to how high I jump.
I ride each of my horses a different amount based on what each individual needs. Though with almost all of mine we'll jump for 10 days straight before each and every show (maybe 5 or so jumps each day....not a lot).
But in terms of regular work, my AO Jumper mare gets 7 rides a week through the show season and we jump close to every day. Mostly we jump 3'6" and under and just a few fences or a gymnastic worked into the flatwork. I've found that it keeps her in better condition for the shows to be kept on a regular schedule like that. She's jumping at the top of her scope, so she needs to stay as uber fit as possible. She also gets two-a-day workouts as often as possible.
My AO Jumper gelding is a super fit TB who jumps the big jumps like they're crossrails. There's no need to work to make him fitter, so he gets 4-5 days of flatwork (lots of dressage/lateral work) and maybe one day of jumping, though often we'll go 2 weeks between jumping sessions. On a somewhat related note, though, he also jumped close to every day for the first year or two of his jumping. Same thing as above....little tiny fences (2'6"-3' for him) and lots of gymnastics. We'd often incorporate a jump into our flatwork and jump it 3 times and be done with the jumping for the day.
My little tiny mare who does the 3'6" jumpers with my nanny gets jumped 2x a week, but primarily so the kid can jump her. The mare herself could probably go without ever seeing a jump between shows.
I HATE HATE HATE the saying that "a horse only has so many jumps in them." That's the same as saying "a person only has so many breaths." Sure....that's true......but you sure can change that number through proper exercise, nutrition, etc. Well, the same goes for a horse. On the one extreme, if you save up all of your jumps for a show you're more likely to end up with a sore or injured horse. On the other, you WILL break your horse down if you school them over 100 jumps a day. But there's a middle ground where jumps can help build the fitness of the horse and maintain/build muscles for the jumping involved in horseshows. It's tough in a boarding barn/trainer-type situation, however, where you need to be in a lesson to jump (or even have lessons regularly), because you do jump over a lot more fences at one time. Because I jump my horses on my own they rarely do more than 10 fences in a day....even an at-height day.
Anyhow, my point is that there is no one answer. Every horse needs something different. Some horses can go without ever jumping between shows, and some need to jump as often as possible. I have both extremes and I treat them very differently. But first and foremost in my mind is the fact that I want them to jump often enough that a horseshow NEVER feels like a strain.