As the fullerene sum formula can be defined as $\ce{C_n}$ with $n$ being any even integer number, is there any fundamental theoretical limit as to how big $n$ can get?
If there is no limit to the chemistry, but there are difficulties in synthesizing large fullerenes, what has been the largest fullerene ever made? Would there be any differences in behaviour?
Not a full answer, but possibly the start of one. I came across a paper[1] discussing some mathematical properties of fullerene graphs. In section 4.2, they give a formula for a lower bound on the diameter based on the number of vertices $n$, assuming there are only pentagonal and hexagonal face. $$\text{diam}(F)\ge\frac{\sqrt{24n-15}}{6}-\frac{1}{2}$$
Given a lower bound on the diameter, it may be possible to prove that a ball of a certain size would be mechanically unstable. I suspect this would give a fairly high upper bound on the size and it wouldn't account for the wide variety of shapes that are possible for fullerenes[2], but this would at least be a starting point.
This could hopefully be refined somewhat by using the largest synthesized/reported fullerene size as a lower bound. Thus far $\ce{C_{500}}$ is the largest fullerene I can find experimental evidence for [3,4]. I suspect this is still pretty far from the largest possible fullerene, even considering just spherical/icosahedral structures.
Based on [2], you will need to look for some kind of mechanical/kinetic stability limit for the upper bound, as the paper suggests that fullerenes should increase in thermodynamic stability as $n$ increases, approaching the structure of graphene or a infinite nanotube.
+200 :-)
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