Divergence of a driftless Brownian exponential clock (${\int }_0^{\infty }{e}^{c{B}_s}ds=\infty \text{almost surely}$)

For any real constant $c$, one-dimensional Brownian motion repeatedly returns to zero. Consider successive segments from zero until exiting $(-1,1)$, separated by returns to zero. Their capped durations are independent identically distributed positive bounded random variables by the Strong Markov property, so their sum diverges by the strong law of large numbers. On each such segment $e^{c{B}_s}\ge e^{-|c|}$. Thus the exponential clock has infinite lifetime, despite arbitrarily deep negative excursions.