TITLE: Categories with some but not all exponentials
QUESTION [6 upvotes]: The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials. 
Are there not-too-esoteric examples of categories of independent interest for which it is significant that they have some exponentials (and more than just trivia like $A^1$) but they don't have all exponentials? 
(For example, are there interesting cases where in effect we can't iterate exponentiation?)

Ooops. I really should not have asked this question as is, since the topological example given by Najib Idrissi is already there (in headline form) in this Wikipedia entry which I indeed recall reading not so very long ago. A senior moment, I fear! But given the helpful pointers to extra detail in  Najib Idrissi's answer, I'll certainly leave this question in place.
But are there also nice non-topological examples, I wonder?

REPLY [2 votes]: For a very small example you can take the well known five-element
nondistributive lattice $ \{0,a,b,c,1\} $ with $ 0<b<a<1 $ and $ 0<c<1 $,
regarded as a category.
Then e.g.  $ b^c=a $ but $ b^a $ does not exist.