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
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