There is an example which seriously stunned me two years ago when I first learned about Hopf algebras. In hindsight it is not that much surprising...

Hopf algebras are usually sold as generalizations of groups. Now, in a group, the notion of "inverse" is something that has to do with one element only: If $g$ is an element of a group $G$, then the inverse of $g$ is defined to be a $g^{-1}\in G$ satisfying $gg^{-1}=g^{-1}g=e$. In contrast, in a Hopf algebra, it is hard to tell whether one given element equals the antipode of another one just by looking at these elements: The axiom for the antipode is 

$S\left(x_{(1)}\right)x_{(2)}=x_{(1)}S\left(x_{(2)}\right)=\varepsilon\left(x\right)$ for all $x$,

and checking this can only be done by checking it for all $x$ simultaneously. When I was a beginner with Hopf algebra, this fact foiled all my attempts at proving elementary properties of the antipode (such that: it is unique, it is an anti algebra homomorphism, any finite-dimensional sub-bialgebra of a Hopf algebra has an antipode, etc.), until I started considering the elements of a Hopf algebra as one big hive rather than a collection of detached things.