It is not difficult to see that the collection of $C^{\infty}$ functions that **fail to be analytic at each point** is $c$-dense in the space of continuous functions defined on a compact interval (sup metric). To see this, let $f$ be such a continuous function and choose $\epsilon > 0.$ Next, pick any $C^{\infty}$ and nowhere analytic function $\phi$ that is bounded between $-1$ and $1.$ (Take $\frac{2}{\pi}$ times the arctangent of an unbounded example, if an example bounded between $-1$ and $1$ isn't handy.) Let $P$ be a polynomial whose sup-metric distance from $f$ is less than $\frac{\epsilon}{3}$ (Weierstrass's Approximation Theorem). Now let $g = \left(\frac{\epsilon}{3}\right)\phi + P.$ Then, for each of the $c$-many real numbers $\delta$ such that $0 < \delta < \frac{\epsilon}{3},$ the function $g + \delta$ is: (a) $C^{\infty}$, (b) nowhere analytic, (c) belongs to the $\epsilon$-ball centered at $f.$ This last part involves the triangle inequality, and earlier we need the fact that if $\phi$ is $C^{\infty}$ and nowhere analytic, then the composition $\arctan \circ \phi$ is $C^{\infty}$ and nowhere analytic and $\left(\frac{\epsilon}{3}\right)\phi + P + \delta$ is $C^{\infty}$ and nowhere analytic. Note that we can also easily get $c$-many such functions arbitrarily close (sup metric) to any continuous function defined on $\mathbb R$ by appropriately splicing together functions on the intervals $...\; [-2,-1],$ $[-1,0],$ $[0,1],$ $[1,2],\; ...$

Any type of cardinality result is pretty much maxed out by this result, but by considering stronger forms of "largeness" we can do better. The results I know about involve Baire category and the idea of prevalance (complement of a Haar null set), and each implies the $c$-dense result above (and much more). Back in 2002 I posted a couple of lengthy essays in sci.math about $C^{\infty}$ and nowhere analytic functions. For some reason they were never archived by google's sci.math site, but they can be found at the Math Forum sci.math site. One day I might LaTeX these essays for posting in this group, but I doubt I'll have time in the near future.

ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS Part 1 (9 May 2002) and Part 2 (19 May 2002)