If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by
$$\Delta(v) = v \otimes 1 + 1 \otimes v.$$
With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation:
$$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$
while the multiplication is given by the shuffle product:
$$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$
where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e.
$$\sigma(1) < \dots < \sigma(k)$$ and
$$\sigma(k+1) < \dots < \sigma(n).$$
I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.