You can't think of any real number. You can think of some of them, but you need to have a way of thinking of them with some specification that can be written down, and those form a countable set. We also have finite life spans, so we actually can only think of finitely many numbers. When you think of the natural numbers, you think of them as a set and concept, which is different than being able to visualize and conceive of every natural number.

What about potentially think of? Even that does not make sense because every possible description you can think of must be written with finitely many symbols in some language, and all such descriptions form a countable set.

Therefore, there are real numbers that could never be written down in English, even potentially. Of course, that partially also depends on the kind of axiom system that you are using. In effect, the standard axioms of mathematics state the existence of things that cannot be effectively specified, which some people actually are against, although such people form a minority.

The surprising thing is that we can effectively reason with large infinities and such objects make intuitive sense (the set of all functions from the natural numbers to the natural numbers is uncountable but a very natural sounding set), and yet it is impossible even in principle to write every one down, even with an unlimited amount of time.