There’s only one player that might be available that the Celtics would consider giving up the top pick for.
In case you didn’t know, the Celtics own the first pick in the 2017 NBA draft thanks to a trade with the Nets in 2013. With that pick, they are likely to select Markelle Fultz.
Fultz is an incredibly talented guard that has the potential to be a perennial all star, and become the scoring partner that Isaiah Thomas needs. The Celtics aren’t like many teams that pick first however. They were in the eastern conference finals just a week ago. So why would they add a 19 year old to a roster one round away from the NBA Finals?
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They certainly don’t have to do that. There will be plenty of suitors that would trade their best player to start over and rebuild around the talents of Fultz. Some examples include Jimmy Butler of the Bulls, Paul George of the Pacers, and Kristaps Porzingis of the Knicks. The Sixers will try to move up to #1, but they’d have to give up more than they’d like for it since they aren’t in a start over stage, they’re more in the middle of the building.
Butler is a fine player don’t get me wrong, but his style of play doesn’t fit the Celtics as well as the others do. If Boston were to give up the first pick, they’d find themselves in NBA purgatory pretty soon because Butler isn’t going to get them over the hump against the Cavs.
George would be a player to get the Celtics over that proverbial hump, but there has been so much talk that he’ll just leave for the Lakers in free agency in 2019, that I’d be very reluctant to give up the first pick for him.
Porzingis is a player that would fit he team and not leave in free agency, but I’m not sure that the Knicks would let him go. Porzingis might want out, but he’s the sole reason that fans will still go to games next season, so I’d presume he won’t be available.

The one player that I’m sure every Celtics fan alive would be comfortable dealing the first pick for is Anthony Davis. The Celtics should give up both of their Nets picks, including the #1, Jae Crowder, and some other piece to get Davis in green.
Davis proved that he is probably the best all around big man in the NBA this past season, yet his team hasn’t surrounded him with a ton of talent. New Orleans did trade for DeMarcus Cousins before the trade deadline, but he might leave in free agency next season if the project continues to not work out.
Davis would be that final piece that Boston has been looking for to take them over the hump and beat the Cavaliers and Golden State Warriors. Not only does Davis fill Boston’s need of a secondary scorer next to Thomas, but he also fills the Celtics’ need of rim protection and rebounding. He average 12 rebounds and 2.2 blocks last season.
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Fultz is going to be a really good player, but Davis is a generational talent that is worth the first pick in the next two drafts. If New Orleans picks up the phone on draft night, Danny Ainge has to give them his best shot at acquiring this insanely talented player that the Pelicans have seemingly wasted so far in his career.