So, even though I said I didn't want a dog, I turned up with Stray Dog. He's nice, but sometimes I'd like him to go chill out somewhere. I can either tell him to wait, which I know is for a limited time, or tell him to go home, which is where he'll go if I make him wait too long.
Thing is, I picked him up when I saved him from a wolf on the road, wandering around. There was no "home" that I saw. So if I tell him to take a break, where the heck will he go so I can find him again?

According to TES Wiki, he/she goes to where the random encounter in which you met him occurred.

Ohh - I hope I meet that Stray Dog some time!

I had Meeko once, but Marcurio killed him during a bandit encounter because he loves his chain lightning... (He tries to kill Shadowmere all the time too, but with no success )

-And the stable guy in Markath won't sell me a dog, he keeps saying I already got an animal :S

So, seems there's one option left - to meet the Stray Dog?

Thanks- I'll try heading back there.

I found like three so far, and they pretty much svck.

I met the stray dog at the worst possible place. There was a wild melee with a hunter, a wolf pack, Atronarchs, AND a random dragon. I just thought that "Stray Dog" was a scripted casualty and didn't realize that he was just an NPC dog in the wrong place at the wrong time.

Stray Dog's a little quirky- he likes to disappear sometimes, when you tell him to wait outside a building/dungeon, or when travelling between areas, he'll sometimes just disappear. Might be a glitch, might be his personality- he IS named Stray Dog, after all....
Anyway, he's better than Barbas, who you get as part of a Daedric quest- he pushes yo whenever you stop walking.
Both of them bark quite a bit, which can get annoying, but it's not super-loud so you can tune it out.

I met Stray Dog while trudging through a blizzard on the way to Winterhold. I heard his howls as he fought a pair of wolves, and thought to myself "this barren wilderness is no place for a dog!" so I asked him to come with me. His company was very welcome, but a few days later when I asked him to wait outside Markarth's Dwemer Museum (I thought it appropriate to keep animals away from such a pristine collection) he disappeared. I have missed him. It's a long way back to Winterhold from Markarth, but maybe I'll find him there again someday.