Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$.
What is the the smallest convex shape this progression of $n$ (or $2 n$) shapes can fit into? For tightly packed squares in rectangles, solutions up to $n=32$ are known (OEIS A081287). No solutions above 32 are known. There are likely tighter solutions if all convex shapes are considered, but these rectangles provide a good starting point. If the outer shape is a square A081287 has some solutions, which were recently corrected for 37 squares.
If you have 1 shape of size 1, 2 shapes of size 2, up to $n$ shapes of size $n$, you have the partridge problem. In this variant, the 30-60-90 triangle has a perfect packing at $n=4$.
For the isosceles triangle set up to $n=11$, I'm fairly sure this is the smallest possible convex shape. What are the best solutions for smaller/larger $n$? (Can you find the perfect packing for $n=3$?)
For dominoes, there is a perfect packing at $n=2$. What is the behavior of tight progressive domino packings as $n$ increases?
Math Magic has other packing sequences. From the no touch tiling page, some other shapes are suggested that might be good at progressive packing, such as the tritan. The below is a tight but flawed solution, the two smallest tritans are omitted, and only one of them has a place to fit. The triamond and tridrafter might also be good shapes.
What are other good shapes to consider? For those shapes, what does the sequence of solutions look like? What shapes have perfect convex packings when $n>4$? I suspect that the tritan and isosceles triangle both lend themselves to perfect solutions.