There are many examples constructed with weighted shifts. The following is a Hilbert space example.

Let us consider the Hilbert space $L^2((0,1),\mu)$, where $\mu$ denotes the measure defined by
$$\mu(A):=2\lambda(A\cap(0\tfrac12))+\lambda(A\cap(\tfrac12,1)).$$
for all Lebesgue measurable sets A. Here $\lambda$
is the Lebesgue-measure. Furthermore, let $T(t)$ be the nilpotent left shift semigroup. Obviously, $T$ satises the semigroup property and, since the norm
$\|\cdot\|_{\mu}$ is equivalent to the norm $\|\cdot\|_{\lambda}$, $T$ is strongly continuous.

Clearly, $\|T(t)\|\leq 2$.

In addition we see that $T(t) = 0$ for all $t>1$.

Finally, consider the function
$$f_t = \frac{1}{\sqrt{t}}\chi_{(\frac12,\frac12+t)}$$
for $t\in(0,\tfrac12)$.
Clearly, $\|f_t\|_{\mu} = 1$ and
$$\|T(t)f_t\|_{\mu} = 2.$$

**ADDED:**

Theoretically, you can always introduce an equivalent norm in your space which makes your semigroup a contraction semigroup, see Lemma II.3.10 in K. Engel, R. Nagel
One-Parameter Semigroups for Linear Evolution Equations. The construction is, however, in most cases highly non-constructive and can only used for theoretical purposes.

It is more important that there are examples in Hilbert spaces where it is imnpossible to find an equivalent *Hilbert space* norm (i.e., the semigroup is not similar to a contraction semigroup), see

Packel, E.W.
A semigroup analogue of Foguel's counterexample. (English)
Proc. Am. Math. Soc. 21, 240-244 (1969).