I think Asik will cost $10-11 million a year on his next deal. So ... I don't think there will be any Big Threes (on max to near max contracts) with a fourth option like Asik earning that much money.
I think it will have to be a different type of roster construct than that.
Say something like (a) the 2004 Pistons where you have a top tier defense, a balanced starting five and one of the best benches in the league. Or (b) something like a Big Four where you have 1-2 players on max or near max contracts and then two players who are below All-Star level earning around $10 million a piece. Sort of like Chris Paul's Hornets when he had David West, Tyson Chandler and Peja Stojakovic (who unfortunately fell off early into their time together). So say Rondo, Kevin Love (2nd max money guy), Omer Asik and 4th guy around Paul Millsap / Lamar Odom / Klay Thompson level performer.
I agree a big 3 isn't likely. I personally would rather see a well build solid team rather then just getting the 3 biggest stars possible and filling in the holes.
I personally will be satisfied with a Rondo, AB, Green, Asik starting lineup for next season.
Ideally I would like to see AB move to the 6th man role and a 3pt shooting vet in the starting lineup. I think he can join KO and an athletic 3/4 (Aminu like) to form a very capable bench.