Denote by $H^\infty$ the Banach algebra of all bounded analytic functions on the unit disc. Then the maximal ideal space of $H^\infty$, (i.e., the set of all norm continuous, multiplicative linear functionals on $H^\infty$) has cardinality $2^{2^{\aleph_0}}$. Since maximal ideal spaces are used all the time in Banach algebra theory, this is an example of a naturally occurring ``large cardinal".
This is also an example of how knowing the (large) cardinality of a set may help direct us in research, even if this knowledge does not lead to a proof of anything. In this example, since the maximal ideal space is so wild, one realizes that it is ``hopeless" to find a nice analytic or geometric description of the maximal ideal space, and one is led to look for a suitable replacement. In algebras such as $H^\infty$ it is often the weak-$*$ continuous multiplicative linear functionals which are more useful (there are only $2^{\aleph_0}$ such functionals).
