Terry Tao has already mentioned Zorn's Lemma in order to find maximal elements in small partial orders. More generally, colimits in categories usually only exist for small index categories. In fact, every category admitting colimits for all index categories is equivalent to a partial order.

Another typical example of this kind is the small object argument. It says that any set of morphisms in a category with certain conditions produces a functorial weak factorization system. The transfinite construction doesn't stop when we start with a class of morphisms.

Another example: A cocomplete symmetric monoidal category is closed if and only if all functors $X \otimes -$ satisfy the solution condition. Todd Trimble has given an example where this fails. It is interesting that being closed is only a property of the data, but the property seems to depend on the size.

Terry Tao has already mentioned Zorn's Lemma in order to find maximal elements in small partial orders. More generally, colimits in categories usually only exist for small index categories. In fact, every category admitting colimits for all index categories is equivalent to a partial order.

Another typical example of this kind is the small object argument. It says that any set of morphisms in a category with certain conditions produces a functorial weak factorization system. The transfinite construction doesn't stop when we start with a class of morphisms.

Another example: A cocomplete symmetric monoidal category is closed if and only if all functors $X \otimes -$ satisfy the solution condition. Todd Trimble has given an example where this fails. It is interesting that being closed is only a property of the data, but the property seems to depend on the size.