It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably infinite. Are there any well known results within say, algebra or analysis that require some given set to be of cardinality strictly greater than $2^{\aleph_{0}}$? Perhaps in a similar vein, are any objects encountered that must have size larger than $2^{\aleph_{0}}$ in order for certain properties to hold?
The Zariski tangent space at any point of a positive dimensional  $C^1$-manifold $X$ has dimension $2^{2^{\aleph_0}}= 2^{\frak c}$.
Let me explain in the case when $X=\mathbb R$.
Consider the ring $C^1_0$ of germs of $C^1$- functions at $0\in \mathbb R$ and its maximal ideal $\frak m $ of germs of functions vanishing at zero. The cotangent space at zero of $\mathbb R $ is $Cot_0=\frak m /\frak m ^2$ and the Zariski tangent space is $T_0=(Cot_0)^{\ast}$ (dual $\mathbb R$-vector space).
Now the germs of the functions $x^\alpha $ are linearly independent modulo $\frak m ^2$ for $\; \alpha\in(1,2)$ . Hence $\dim_{\mathbb R} (Cot_0)=\frak c$ and so indeed the Zariski tangent space at zero of $\mathbb R$ is $\dim_{\mathbb R} (T_0)=2^{\frak c}$.
It is noteworthy that many textbooks erroneously claim that for an $n$-dimensional manifold of class $C^1$ the Zariski tangent space defined above has dimension $n$. Or they make some equivalent mistake like claiming that the vector space of derivations of $C^1_0$ has dimension $n$ . An example of such an error is on page 42 in Claire Voisin's (excellent!) book  Hodge Theory And Complex Algebraic Geometry I published by Cambridge University Press.
To end on a positive note, the phenomenon I am describing only raises
its ugly head for $C^k$-manifolds with $k<\infty$. For $n$-dimensional $C^\infty$-manifolds the Zariski tangent space at any point has dimension $n$, as it should. The heart of the matter is that a $C^\infty$ function $f$ ,
on $\mathbb R$ say, which vanishes at zero can be written $f=xg$ for some function $g$
which is also of class $C^\infty$, whereas $g$ would only be of class $C^{k-1}$ if $f$ were  of class $C^k$.
