I've been questing, killing rares, and collecting treasure for just a day now on about 3 characters, collecting many items to upgrade my Artifact Power.
Above the experience bar is another bar showing the current and total Artifact Power I have collected for my weapon:
At level 100 it was a maximum of 300.
After leveling to 101 and upgrading my weapon once it has a new maximum of 325.
However I realized, that is not the actual maximum.
When approaching the 300 mark, I was at 265 and clicked a +100 Artifact Power item, which put me at 300. I was a bit dissapointed that I just wasted 65 power, but after upgrading my weapon seconds later the bar reset to 65 / 325 power.
This means that my 65 power was not wasted, and the number shown on the bar is not the maximum.
That all being said, is there a hard maximum amount of power your artifact can store without going to your Order Hall and upgrading it?