MATHS!
If the Big Apple apple has a volume of
x and each small apple has a volume
y, then the upper limit of the number of apples (assuming you can change their shape) would be
x/y. However, intuition tells us that the actual number will be less than that, as there will be voids in the space.
This sort of lattice packing is actually incredibly important for many applications in materials science (it's why diamond is so strong, but graphite sides apart easily, even though they're both pure carbon - it's all to do with their lattice structure and bonds). As it's so important, a lot of people spend a lot of time (me included for a handful of modules at university), studying them and understanding how they work.
A close packed structure (either face centred cubic [FCC], or hexagonal close packed [HCP]) are the densest way to pack spheres. There's an upper bound, which can be proved with some maths, that says of a given volume the most space that you can fill up with spheres is about 74%. The rest of that space is unoccupied.
Let's assume the Big Apple apple, and all of the real apples, are spheres and plug the numbers in.
Let's assume the average apple is 8 cm in diameter. It's volume, therefore, is (4/3)*pi*0.04^3 = 0.000268 m^3
Let's also assume the average Big Apple apple is 3.5 m in diameter. It's volume, therefore, is (4/3)*pi*1.75^3 = 22.45 m^3.
However only 74% of that volume is occupied by apples (the rest is air), so we need to fill 22.45*0.74 = 16.61 m^3 with apples.
16.61/0.000268 = 61977. I'll round that to about 62000 apples.
Sounds like a hell of a lot - which it is. In reality, you'd probably get a packing density of more like 50%, which is about 42000 apples. And it's worth remembering that neither the Big Apple apple, nor the real apples are likely to be spherical. It still seems like a number much higher than you'd guess, but the maths is there (unless I've made a blunder)!
EDIT: For clarity.
I'm talking about this kind of Big Apple:
Not the ones that have just a smaller one at the top of the lift hill like this:
But you can just repeat the process for any size Big Apple (or apple) you like.
