Let's make a few assumptions:
1. Ray Allen won't leave for a non-contender
2. Ray Allen won't sign for the MLE or less
These may not be right, but I think Boston will offer him more than the MLE and a contender, so I think that's what it takes to beat BOS' offer.
Candidates to offer more than the MLE w/o a S&T:
NYK, NJN, LAC, MIA, CHI... not sure who else.
LAC could make a deal with Gordon + others to get a SF and then bring in Ray at SG, but it's not likely.
NYK, NJN - probably only if they sign Lebron, I don't think Knicks can sign Lebron and beat an MLE type offer to Ray. They will have to renounce David Lee. I don't think Lebron is going to Nets, but if he does, I could see them making a strong play for Ray and moving their current back court. They could also keep Lee and Harris who might be nice compliments to Lebron anyways. Too many dominoes have to fall on this one, I'm not scared.
MIA - kind of makes sense, let's say they trade Beasley for a true center, sign a couple of minimum guys to fill out the roster and bring in Ray. I think that team still competes. But better to go after a guy like Miller who could give you 5 years instead of 2-3 from Ray. Miller also has the size to switch between SF/SG which will give you some roster flexibility. Additionally gives you the LLE and MLE to fill out the roster for a run in 2012, whereas with Ray you'd want to be in position to make a run right now.
CHI - this one makes the most sense, I think they can land Boozer for less than the max, maybe only a four year deal too. (I'm guessing the chairs are filled when the music stops for him.) Then you sign Ray for a three year deal. That gives you a chance to compete now, Ray and Boozer coming off the books as you start having to pay more for Noah and Rose, and the chance to keep building around what you have. Noah, Boozer, Deng, Ray, Rose is a very nice put together starting 5. [They need to fill out the bench with a trifecta of PJ Brown, Posey, and House to complete the set though

] Plus Ray is comfortable with Thibs, comfortable paying with a PG that controls the ball. I see this as biggest threat.
Dark horse would be CLE just offering the full MLE and Ray getting p---ed that Danny won't go three years, in this situation, Ray might bounce. A max three/four year MLE with max raises gets him through his career and makes as much money as a two year here. And he get's to play with the perfect compliment to his game in Lebron. Again unlikely he takes that pay cut short term though (since he'd be giving up ~4 mill in year one.)