I will elaborate on Bruce's examples and add a couple of my own.

Of course, the homology and cohomology of topological groups over a field are good examples.

For each prime prime $p$ the Steenrod algebra $\mathcal{A}_p$ which is the algebra of endomorphisms of the cohomology theory $H^*(-;\mathbb{F}_p)$. The cohomology of this Hopf algebra is the $E_2$ term of a spectral sequence, due to Adams, converging to the $p$-completed stable homotopy groups of spheres.

The functions on any affine algebraic groups over a field are another family of examples.

Formal group laws over a field $k$. You can read about these in Husemoller's book.

The rational homotopy groups of connected topological group or more generally an $H$-space is a Lie algebra. A nice result of Milnor-Moore shows that the universal enveloping algebra of this Lie algebra is isomorphic as Hopf algebras to the rational homology of the space.

Of course, the homology and cohomology of topological groups over a field are good examples.

For each prime prime $p$ the Steenrod algebra $\mathcal{A}_p$ which is the algebra of endomorphisms of the cohomology theory $H^*(-;\mathbb{F}_p)$. The cohomology of this Hopf algebra is the $E_2$ term of a spectral sequence, due to Adams, converging to the $p$-completed stable homotopy groups of spheres.

The functions on any affine algebraic groups over a field are another family of examples.

Formal group laws over a field $k$. You can read about these in Husemoller's book.

The rational homotopy groups of connected topological group or more generally an $H$-space is a Lie algebra. A nice result of Milnor-Moore shows that the universal enveloping algebra of this Lie algebra is isomorphic as Hopf algebras to the rational homology of the space.