TITLE: A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?
QUESTION [5 upvotes]: I'm interested in finding an example of a locally small category $\mathcal{C}$ having

small filtered colimits and
arbitrary small products

but lacking, either

all small limits,

or either

the property of finite limits commuting with filtered colimits.

If I'm not wrong, any decent forgetful functor $ \mathcal{C} \longrightarrow \mathbf{Sets}$ from my category $\mathcal{C}$ to the category of sets, would make $\mathcal{C}$ fulfill properties (1) and (2). Hence, I have to discard all the usual concrete categories, haven't I?
I would be very grateful of any counterexamples (even the most obvious and embarrassing ones :-)  ) anyone can tell me.
EDIT. Dear Zhen Lin and Hanno: thank you both for your examples. They are really useful to me.

REPLY [5 votes]: The category of Kan complexes is a full subcategory of the category of simplicial sets that is closed under filtered colimits and small products, but it fails to have equalisers.
Consider the ordinal $\omega  + 1 = \{ 0, 1, 2, \ldots, \omega \}$. Let $\mathcal{C}$ be the poset of $\sup$-closed subsets of $\omega + 1$. It is not hard to see that $\mathcal{C}$ is a complete lattice, so it is also cocomplete. Let $A = \{ 0, \omega \} \subset \omega + 1$ and let $B_n = \{ 0, \ldots, n \} \subset \omega + 1$. Clearly, the colimit of the chain
$$B_0 \to B_1 \to B_2 \to B_3 \to \cdots$$
is just $\omega + 1$ itself. On the other hand, the colimit of
$$A \cap B_0 \to A \cap B_1 \to A \cap B_2 \to A \cap B_3 \to \cdots$$
is $\{ 0 \}$. Thus, finite limits fail to commute with filtered colimits in $\mathcal{C}$.