TITLE: Do numbers get worse than transcendental?
QUESTION [89 upvotes]: Mathematicians have come up with many ways of classifying how "exotic" some numbers are. For example, the most ordinary numbers are the natural "counting" number, and the next most exotic numbers are zero and the negative integers. Next are the rational non-integers, which are ratios of integers. Then come the algebraic numbers, which can be expressed as roots of polynomials with rational coefficients. But there are still transcendental numbers, like $\pi$ and $e$, which cannot be expressed in any of these ways. However, $\pi$ and $e$ can both be expressed as integrals of elementary functions.
My question is this: Do there exist any numbers (transcendental, of course) that cannot be expressed as the definite integral of an elementary function? Of course, I don't expect anyone to prove something like this (perhaps disprove it?), but I would like to know whether something like this is likely. Clearly, all real numbers can be expressed as an infinite series (just by using egyptian fractions and the greedy algorithm), but whether all real numbers can be expressed as an integral is less clear.
My thoughts: I really have no idea, but if such a number did exist, I would expect it to be something with infinite irrationality measure (like Liouville's constant, or any other Liouville number).
EDIT: To clarify, my question isn't really about "what other types of pathological numbers are there". I really want to know if anyone can provide any computable examples of numbers that aren't exponential periods.

REPLY [8 votes]: Do there exist any numbers (transcendental, of course) that cannot be
  expressed as the definite integral of an elementary function?

The answer is simply no because, as Jack D'Aurizio pointed out in the comments, any real number $x$ can be written as
$$
x = \int_0^x 1 \, dt.
$$
Alternatively, we could write
$$
x = \int_0^1 x \, dt.
$$