The Celtics are not going to have the cap space to offer anybody a max deal. Assuming they pick up Sullinger's option and bring back Avery on either the qualifying offer or a new deal, they should only have around 10 million if the salary cap rises to 61 million. Wallace's contract ate up much of the projected cap space but given the draft picks they got and how they aren't going to be an attractive free agent destination anyway, it's not a big deal.
with cap holds, rookie contracts, etc. they will have a lot less than that. Boston only will have at best 5 million in cap space, or essentially the MLE. In other words, no cap space.
The Utah Jazz received two two first-round picks and three second-round picks as compensation for absorbing the contracts of Andris Biedrins, Richard Jefferson, and Brandon Rush.
It should take something less than that to find a team with cap space who would to be willing to accept a pure salary dump of the Brandon Bass (expiring) and Courtney Lee (not expiring), which would vault the Celtics from $5m (if that's what they have) in cap space to over $15m.
They could also offer the expiring contract of Bass and the unguaranteed contract of Bogans as trade ballast if they can get another team to agree to a sign-and-trade of a free agent.
If a max contract free agent wants to come to Boston, it shouldn't take more than two first-round picks, and might take less, to make that happen.