Why do the speed settings on fans always go [OFF -> HIGH -> MED -> LOW] instead of [OFF -> LOW -> MED -> HIGH]?

The fans I’ve seen are “Off, Low, Med, High.”

You most often see this in fans that have a single switch that gets moved from setting to setting. The short answer for why is “efficiency.” It usually takes more power to start a fan from a dead stop than it does to keep it running at Low, so making Low the first setting could result in a slow start when you turn the fan on. By making High the first setting, you get the biggest jolt from the start capacitor. This gets the fan running at speed more quickly no matter which setting you switch it to after turning it on. It also means that the fan benefits from the extra momentum if you switch it over to Medium or Low after starting it up. This last benefit would be pretty small, but every little bit of efficiency counts to a good engineer.

The older the fan is, the more this efficiency matters. An old fan that uses Low as its first setting is likely to have more and more trouble starting up as it ages. So if you’re hoping to get the most out of your fan, you’ll want one that goes from Off to High.

For handicapped remote people who are already to tired from getting up from the couch to turn the dial.

To expand on @JeSuisRickSpringfield‘s already excellent response, there may be some minor electrical engineering involved here, too. (As well as sales and marketing.)

The first two positions of the switch are the binary ones for “all off” and “all on” (that is, “full power / high speed”), then followed by the step-down switches for lower speed modes. I’m going to hazard a guess as a non-electrical engineer that those are the two positions of most stress for the electrical switch, too, so it makes sense to juxtapose them and shield them equally well and together, so that the less-used and less electrically stressed switches (and this is where I think the sales and marketing data become helpful, so they know this) can be offset as they are.

I would also imagine that the sales / marketing people have researched the market to know that when most customers turn on a fan they normally want it “full on”, and then only later want to reduce power.

One tiny addition to @JeSuisRickSpringfield answer.
By hitting the fan motor with full current at start up you reduce the chance of the fan being stuck in the stalled position.
If the fan has not been run in a long time and dust or rust has gotten inside it might take more juice to get it to spin than the “Low” setting can supply. In that case the fan would just sit in one spot as the motor got hot and, in the absence of air cooling, eventually damaged itself.
Fans with digital controls (from this century) know this and can be placed in any order.

My extractor fan for my bathroom is like this. Now I know why! Thank you!

The switch for my attic fan is similar, except that it has a dial. Turn the switch, it’s on “high”. Then,after it’s running fully we can lower the speed bt turning the dial.