Consider the category of $k$-algebras with $k$-algebra homomorphisms. It's clear that the field $k$ itself is the initial object, since any $k$-algebra morphism must fix $k$.
Do terminal objects exist in this category? I looked all over and couldn't find mention of one. Any terminal object would have to contain all of $k$, but I feel there should be other $k$-algebras out there which have some independent generators over $k$, and then by choosing where to send the generators, you could construct multiple $k$-algebra homomorphisms into any other $k$-algebra, so a terminal object can't exist.
Am I over looking something?