If you have the financial ability to travel where ever you want, is there always a ski resort with snow ready for skiing or snow boarding available somewhere in the world at every point in the year?
Looks like it: Summer Skiing on Grass
Yes. That isn’t much of a question at all. South America in places like Chili have opposite winters from North America and high mountains with good skiing. If you want to push the issue, even Hawaii has natural downhill skiing on at least one high mountain. There is no point in the year where there aren’t good ski resorts open somewhere. Even in the U.S. and Canada, some places have very short summers. High peaks in Alaska are one but ski reports in Vermont are open from October until late April.
There’s always Ski Dubai. Not completely natural though. 
You can ski year round in several places without having recourse to artificial snow-making. Folks I knew in Utah used to ski in the summer by gluing “skins” on the bases of their skis, herringboning their way up the still snow-covered peaks, then skiing down. It’s not exactly what you’re talking about, but I note that the ski resorts are open all year, anyway.
Looks like you might not need to travel to that exotic a place: Hintertux Glacier (Austria)
There are several places in Europe where you can ski 365 days a year. Tignes in France is another. Or at least, on Googling, I find that it used to be. Apparently they didn’t sell enough tickets in autumn to make it worthwhile staying open, so it’s no longer open all year.
Ski Antarctica! Never a worry about the lines being too long.
Are there mountains in Antarctica?
Also to the OP. Watch a little Warren Miller and see his documentaries of people skiing everywhere in every season. Mid-August skiing in Chile is amazing. Chest deep powder etc, etc…
Of course? Haven’t you read your Lovecraft? The Mountains of Madness are there!
There are mountains in Antarctica, in any case. I don’t know if there’s anything skiable on them. I’d be surprised if there weren’t, but I’m also surprised I’ve never heard of anyone doing it, just for the hell of it.
http://seven-summits-quest.com/VINSON%202006.htm
Looks like ski-able terrain. Also sounds like a bit of a pig to get to without a lot of logistic support.
I’m pretty sure there are plenty of mountains with snow year round in the US. Mt. Hood, in Oregon, for instance. I don’t know if the ski resorts there are open 365, but it seems likely that you could find it somewhere in just one place in the world-- no traveling required.
In news somewhat closer to home, it’s almost mid-summer here, and the huge mound of snow from last winter, which the city dumped in a snow storage yard near Kipling subway station, still hasn’t melted completely. Of course, it’s covered in dirt and garbage by now, so the skiing probably wouldn’t be all that great.
Year round skiing isn’t that difficult. We have glaciers in the US so backcountry is available all the time. (When you put skins on your skis you don’t need to herringbone; you just ski on AT or Telemark gear) Mt Hood operates nearly a full year for lift served skiing.
There are plenty of ski areas in South America which overlap with US ski seasons so you’d only have to take two flights a year.
I am not so sure about that remark, the resorts down this way (south America) open in the 2nd week of June in a good year and close mid October.
Are there any North American resorts that stay open until mid June or open before the middle of October? If not there may be a a few weeks either side of the respective winters, although the Caribbean is on the way and would make a nice extended stop over.
Arapaho Basin typically operates from late October to late June or early July. Many years Timberline will be open in the Fall, so there might be a week or two without overlap.
And then there was the season when we all stored some snow in our freezers, then padked it up in barrel packs to a ski hill for a 1 July Canada Day ski BBQ.
Timberline Lodge offers year-round skiing, but not 365 days. It’s closed for maintenance for two weeks in September.
Some of the other grad students went up to one of the local mountains to watch the fireworks on July 4, and then skied down. And Bridger Bowl (the most popular resort hereabouts) have run their lifts as late as June 20th.