Hi,

As remarked by Leonid Kovalev a Lévy process doesn't explodes as far as I know, nevertheless as you mention generic Levy Process for which I don't know any definition, you may be thinking of them as diffusions driven by a Lévy process.

Then looking at some SDEs, you can have some cases where explosion time is a.s. finite, and even when the driving procsess is a Brownian Motion, for example I think I can remember that $X_t$ verifying $d[Ln(X_t)]=a(b-Ln(X_t))dt+\sigma.dW_t$, $X_0=0$ is of this type.

(for references, google at "Black-Karasinski short rate model")

Regards