For a field $k$ and a natural number $n$, the vector space of dimension $n$ over $k$ is unique up to a non-unique isomorphism, though this somehow feels "less unique" to me than your other examples. I thought at first that this might be due to its not fitting into the class of examples described by Qiaochu, but I suppose you can force it into that class by considering the category of $n$-dimensional vector spaces over $k$. But that in turn feels considerably more ad hoc (at least to me) than considering the category of algebraic field extensions.
For a field $k$ and a natural number $n$, the vector space of dimension $n$ over $k$ is unique up to a non-unique isomorphism, though this somehow feels "less unique" to me than your other examples. I thought at first that this might be due to its not fitting into the class of examples described by Qiaochu, but I suppose you can force it into that class by considering the category of $n$-dimensional vector spaces over $k$. But that in turn feels considerably more ad hoc than considering the category of algebraic field extensions.