First off, one thing I'd recommend is reading Filinski's Linear Continuations for ideas on how to interpret linear connectives (note, the ? modality got typeset as Γ in that for some reason).
In that paper, he uses the modality as part of the interpretation of call-by-name into linear logic. The idea is that you can kind of think of the non-modal types $T$ as being total, while $?T$ adds the possibility of divergence. If you want an analogue of "infinitely many copies of $T$", then it's, "zero or one $T$".
But of course, it's not like $1 + T$. It's more like a partiality monad living in an otherwise total language.