  We show that a Brownian motion on $\mathbb{R}_{\ge 0}$ which is allowed to
spend a total of $s > 0$ time units outside a bounded interval does not leave
the interval at all. This can be seen as an extreme example of entropic
repulsion. Moreover, we explicitly determine the exact asymptotic behaviour of
the probability that a Brownian motion on $[0,T]$ spends limited time outside a
bounded interval, as $T \to \infty$.
