In our league this has occured and I can't really give an explanation on how the player ended up in AAA.

Team A - Waives and DFAs a player after the transaction deadline
Team B - claims to prevent him from getting to AAA
Team A - does not trade, cannot send to AAA and can't put the player back on MLB
Player's DFA expires and is released
Team B - claims and gets player
Team B - assigns player AAA

How is team B allowed to assign the player to AAA? he is out of options, never cleared waivers and has an ML contract.

Any explanations? 

Once the player is released, he can be assigned to the minors.