What real numbers will we never know (even the approximate) value of?

\tan\left(9^{9^{9^9}}\right).

I contend that we won’t ever know whether this number is positive or negative, whether it is more than a billion or less than a billionth, whether its integer value is odd or even, anything of the sort. It’s a well-defined real number, but to determine any of those things you will likely need to spend more than the energy resources of the entire universe.

(EDIT: To clarify, there are certainly things we know about this number - for example, it is a real number, rather than undefined; it is irrational, and in fact transcendental. I’m specifically answering the question about knowing its ”approximate value”. We can’t place this number in any reasonable interval. Also to clarify, this is the standard \tan function, whose input is expressed in radians).

It is possible, of course, that there’s some insane shortcut, but I wouldn’t expect there to be.

You can easily make up such numbers. This particular one is based on the fact that \tan(x) is very sensitive to the precise value of x \bmod \pi, which means you have to fit in exactly as many \pi's as possible in 9^{9^{9^9}} and know what’s left over with some reasonable precision. That’s unlikely to ever be feasible, and if by some wild miracle it does become feasible, just add one more 9 to the power tower.

Oh, for those of you who are rushing over to Wolfram|Alpha to try this out: let me save you the trouble.

But again, you can concoct any number of such expressions, for whatever that’s worth.