a marble is 12mm diameter. What is the smallest height box that is 96mm square that can hold 200 marbles and still have the lid fit on the box without inference

Answer 1

Option 1-Lay 64 marbles in the bottom of the box forming an 8x8 layer.
Lay a 7x7 layer atop the bottom set where each marble in the 2nd layer nests atop 3 marbles of the lower layer. Lay the 3rd 8x8 layer atop the 2nd layer. This places 177 marbles in the box. The vertical distance between the marble centers is 9.7979mm.
The height of these three layers is 31.5958mm. With 23 marbles left to be accommodated, the box must be 41.3937mm high allowing the remaining 23 marbles to nest atop 3 marbles below.

Option 2-Lay 64 marbles in the bottom of the box forming an 8x8 layer. Lay 7 marbles atop each of the strings of 8 on the bottom, each marble contacting two marbles in the first layer. Lay another layer of 8x8 marbles atop the 2nd layer. This results in 8 rows of 184 marbles with 16 marbles left. These 16 marbles can nest in sets of 3 as in option 1. The total height required for the box becomes 6 + 10.392(2) + 9.7979 + 6 = 42.58mm.