Easy functions? Let $f$ be an analytic function, and suppose that we want to compute
$f(x)$. The input consists of the digits of $x$ and the output of
a rational number approximating $f(x)$. A function $f$ is called easy if
there is
an algorithm which computes $f(x)$ with accuracy $2^{-n}$
using $n^{1+o(1)}$
arithmetic operations.
It is known that elementary functions like $e^x,\log x$ are easy.

Is it known (proven) about any reasonable function
  that it is hard (not easy)?

For an algorithm, using the AGM, showing that $e^x$ is easy, a reference is
D. Newman, Rational approximation versus fast computer methods,
Lectures on approximation and value distribution, pp. 149.174,
Sém. Math. Sup., 79, Presses Univ. Montréal, Montreal, Que., 1982.
EDIT1: The same paper contains a proof that multiplication is easy (fast multiplication), 
and if $f$ is easy then
the inverse function is easy (Newton's method).
EDIT2: I understand that with our present knowledge we cannot compute Euler's constant
efficiently. But I don't know a proof that this is impossible.
Remark. I am mostly interested in analytic functions, even "special functions". Are they all easy?
 A: Any (uniformly) polynomial-time computable function must have a polynomial modulus of (uniform) continuity [Ker-I Ko 1991,Theorem 2.10]. 
The function $0\lt x\mapsto1/\ln(e/x)$ is well-defined on [0;1] and (exponential-time) computable yet has no polynomial modulus of continuity at 0; see Example 1.12 in arXiv:1211.4974. 
It is not analytic at 0, though...
For the stronger question on 'simple' real numbers (i.e. constant functions, cmp. Norbert Müller's answer) that are not computable within polynomial space, say, confer periods in Model Theory and this article by Tent and Ziegler.
A: If we consider constant functions (trivially analytic...), then we could change the original question to: Are there "reasonable" real numbers that are not computable in quadratic time?
As there is a time hierachy theorem on the real numbers (Norbert Th. Müller: Subpolynomial Complexity Classes of Real Functions and Real Numbers. ICALP 1986: 284-293), there exist numbers in qubic time that are not in quadratic time. Whether there numbers are as "reasonable" as $\pi$ or $e$, however, is another question...
A: Here are a couple of relevant references. I will try to find one that is specifically about the exponential functions. But in general it is a bit too optimistic to hope for such low complexity as $n^{1 + o(1)}$. You cannot expect it to be faster than multiplication.


*

*Ker I-Ko, Complexity theory of real functions. Birkhäuser, 1991.

*Klaus Weihrauch, Computational complexity on computable metric spaces, Mathematical Logic Quarterly, Volume 49, Issue 1, pages 3–21, January 2003

