The question is the same as the title. If a more concrete example is required, consider the norm induced on $GL(n,\mathbb{R})$ by any vector norm on $\mathbb{R}^n$. If $A, B \in GL(n,\mathbb{R})$ the only case I can think of where $\|AB\| = \|A\|\|B\|$ is $A = B = I$. Are there any others?
Edit: Considering the comments have brought about some good points regarding the possible scope, let me include the additional questions:
(1) If $\| A\| = \| B \| = 1$, when is it true that $\| AB \| = 1$? This is obviously true for the identity and any unitary or rotational operator.
(2) If $\| A \| > 1$ and $\| B \| > 1$, when is it true that $\| AB \| > 1$?
@JMoravitz will now note that the TeX has been typeset in the One True Holy Kosher Way ;-P