Intuition for not-so-smooth manifolds in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds.
However, neither in these books nor in lectures I have encountered real examples when a manifold may be $C^k$, but not $C^{k+1}$.
Intuitively, I would suppose the $|\cdot|_\infty$-Ball with radius $1$ to be a merely continuous, non-smooth manifold, because smoothness fails at the edges of the cube. In contrast to this, polar coordinates show the $|\cdot|_{2}$ with radius $1$ is in fact a smooth manifold.
I'd be thankful for some examples with clues to basic techniques, how the different degrees of smoothnesses manifest 'in real life'.

Take a look at Morris Hirsch's text "Differential Topology", in particular Theorem 2.9 on page 51.  The upshot is that if $M$ is a $C^k$-manifold for $k \geq 1$, then it admits the structure of a $C^j$-manifold for any $j \geq k$, and that structure is unique up to diffeomorphism.   So in that sense the different degrees of smoothness is largely just an artefact of how your manifold is constructed -- you can always "do better" provided the manifold is at least $C^1$. 
I finished this as Emerton's answer appeared, anyhow, take this as a reference for the result. 
