I've got a late round 5, 2 late round 6ths, and a 7th that I need to move on from.

I will bundle as a three for one to move up slightly up into 6th (if you want 2 sixth rounds and a seventh), or for one to move up slightly in the 5th (if you want a 5, 6, and 7th).

I wonder how many iterations of this kind of trade before it becomes an early 1st round pick.

late 5 + 6,7 -> early 5
early 5 + 6,7 -> late 4
late 4 + 5,6,7 -> early 4
early 4 + 5,6,7 -> late 3
late 3 + 4,5,6,7 -> early 3
early 3 + 4,5,6,7 -> late 2
late 2 + 3,4,5,6,7 -> early 2
early 2 + 3,4,5,6,7 -> late 1
late 1 + 2,3,4,5,6,7 -> early 1

Total paperclips/picks given up that you owned originally
1st: 0
2nd: 1
3rd: 2
4th: 5
5th: 8
6th: 9
7th: 9

2 things:

You miss 100% of shots you don't take...

and

Let's make a deal.