Is it possible to jump multiple times when jumping off an object you are holding? Say you are holding a solid box, and you hold it down below your feet. Then, you jump off of it. Since you are holding the box as well as being boosted up, the box is also moving up with you. Could you then keep repeating this and jump many times off the box?
 A: No, the box will not get the same boost as your body.
The "boost" you are talking about is what we technically call momentum. To jump up you gain momentum, and that momentum was taken from the box. As you jumped up, the box is pushed down. This is Newton's 3rd law. Momentum is conserved. If you then, after having jumped from the box, pull the box up with your arms, then that box gains upwards momentum that your body loses. So you won't move anywhere together (your centre-of-mass will remain stationary).
Now, you can experience lifting the box up along with your body in a jump when you do the experiment as you described. Meaning, you can move the centre-of-mass of you-and-box up in your example. That is because your downwards pushing force actually propagates through the box and into the ground. The ground is pushed downwards in order for you-and-box to move upwards - you-and-box gained momentum and the ground lost momentum (gained downwards momentum).
You won't be able to do a second in-air jump now that you have no ground to jump off from again.
