This thread raises two questions.
What qualifies as a big three?
Answer: I would say 3 year in year out all stars, also probably considered top 5 at their position, and worthy of being paid the max or near max. These are the names that come to mind when one thinks big three:
Bird, Parish, McHale - Garnett, Pierce, Allen - James, Bosh, Wade
Which current teams have a big three?
Answer: none
The Clippers might come the closest. Paul and Griffin are arguably the best their their positions. And Jordan is at least a top 10 center. But he isn't top 5, and has not made multiple all star games or even one all star game. The same goes for Durant, Westbrook and Ibaka. Or Bosh, Wade, Deng. Or Parker, Duncan, Leonard.
The league has a lot contenders with 2 perennial all stars, not 3. Three is very hard to get and very expensive to maintain.