They have a MLE and four TPEs: 17.2m, 9.7m, 6.9m and 3.8m. So they have the chance to add a bunch of significant contributors and remake that bench if they want to.
My guess is they use the big TPE to acquire a shooter -- someone who works off the ball, and that other teams will have to chase around screens every second he's on the court. Duncan Robinson fits but is too expensive. Luke Kennard is ideal, and given that he's no better than 9th on the Clippers depth chart, and completely redundant with Powell on the roster, and that the Clips are massively into both the luxury tax and repeater tax, my guess is he'd be get-able for a future first.
Getting Kennard (or someone in that salary range) would put the Celtics way, way into the tax, and Wyc's not going to be like Ballmer or Lacob in that way, not even with a contender. So my guess is that Nesmith (who I still have stock in) and Theis would both be traded out into someone's cap space. That'd cost second-rounders, most likely, but would also open up two decent-sized (4m and 9m) TPEs they could roll into NEXT summer.
Replacing Theis, who was mostly unplayable in the postseason, would be easy -- get someone like Biyombo. Then the other piece the team would need is an explosive ball-handler who can put pressure on the rim. Devon Carter would be a good example.
At that point, this is the 11-man depth chart that'd put the Celtics into the tax by about $7m:
PG: Smart, White, Carter
SG: Brown, Kennard, Pritchard
SF: Tatum
PF: Horford, Grant
C: Timelord, Biyombo
I don't know if that's going to be good enough to get over the hump -- the Nets, Dubs, Bucks, Clips and Nuggets all have potential to be absolutely massive next year. Bet it's a more talented and explosive group than this year, especially if the guys who matter most continue on their linear improvement.