Here's the actual deal.
You can bank AAs yourself, up to twice your level. No way to bank more than that.
EXCEPT: refunds. Refunds can bring your unspent bank as high as needed. When you next log in, you have to spend them down, so that's why the AA window opens automatically.
Now, here's the one catch: If you're F2P, you can't likely spend them down, because you're already above the 1000 (250?) AA limit. However, the system still lets you spend them anyway, down to 3 times your level. At that point, you can't spend any more, but you won't lose any (if you don't spend them down to under 3 times your level the first time you log in, you will lose them). So, you now have less than 3x your level, so you're safe, but more than 2x your level, which is the normal AA bank limit, so, EVERY SINGLE TIME you log in, the AA window will open, even though you can't spend any.
Tatanka Wolfdancer, 110 Wood Elf Druid, 9 x 300+ Master Artisan, 7 maxed trophies (dang research! :)
Anaceup Mysleeves, 105 Erudite Mage, 2 x 300 Master Artisan
Snookims Whinslow, 105 Erudite Enchanter, 2 x 300 Master Artisan
<Inisfree>, Tunare (Seventh Hammer!)
Tatankaa (95 Inquis)/Urkell (38 Conj)/ Seanthornton (24 Pally), <Inisfree> Freeport server (some other server now)