Depnds what you call "outgrow".
If it means "you'll stop watching 10 episodes per day", then I think most people will go through this. If it means "stop watching anime", it's just not possible.
Like a few others in this thread, I'm in my early thirties. Some years ago I was still crazy watching lots of stuff, including 150-ish episodes of One piece, Naruto and the likes. I would never do that anymore. My taste have changed.
Mind you, it's not a problem of length. I'm watching Legend of Galactic Heroes atm.
It's a problem of taste, + time/value ratio. I still follow Naruto and One piece manga, though. But the anime, no thanks.
What will be close to imposible to stop watching as anime are:
- Good movies (Ghibli, etc)
- Good OVAs or original TV series (GITS SAC for example)
- TV series that bring a sort of new life to the manga which it is adapted from (quite rare but they exist)
That probably does not cover most of the series which are popular at the moment. I would be more inclined to read the manga for most of them.
Besides, it's been a while since I started going back to the origins, watching and reading old stuff. I watched Gundam (original series) recently, I'm about to go into Yamato and other Leiji Matsumoto works, I try to read as much Tezuka as I can.
When you grow old, your "manga/anime experience" also grows. You get tired of the same old stuff. And the beauty of it all is that at that point you have more time to (re-)discover fantastic classics, or to dwell into "darker" corners of the manga maze.
I guess you can say that you outgrow a certain category of anime.