Anvil of doom, can it join units?

To summarize (for future reference):

Q. Anvil of Doom, can it join units?
A. In theory: yes; in practice: no.

BRB p. 97 specifies: "A character is normally permitted to join units of infantry, beasts, cavalry, monstrous infantry, monstrous beasts or monstrous cavalry," with the following exceptions: "Unless otherwise stated, a character cannot join a unit of monsters ..., a unit of flyers ..., a unit of chariots ..., a unit of swarms ... or a war machine. Similarly, a character that is itself a monster, or riding a monster or a chariot cannot join other units." 

While characters cannot join war machines, there is no stated exception prohibiting a character with the troop type war machine to join one of the permitted units. Similarly, a character with (or without) the Fly special rule cannot join a unit with the Fly special rule, but there is no such restriction for a character with the Fly special rule joining a unit without. 

However, BRB p. 5 specifies: ""A unit consists of 1 or more models that are arranged in orthogonal base contact with each other, which is a fancy way of saying 'edge-to-edge and front corner to front corner," which is confirmed by BRB p. 99: "If a character cannot be placed within the unit, or beside the front rank, then he cannot attempt to join that unit. This is obvious, perhaps, but worth saying nonetheless." 

"A model should always be mounted on the base it is supplied with" (BRB p. 80), and the AoD models GW has produced have either no base or a round base. Hence, it is impossible for them to be placed in accordance with the rules, and the AoD cannot therefore attempt to join a unit. 

Nevertheless, as always: if your opponent and you agree, feel free to play it differently.