Which animal never stops growing?

Lobsters keep growing throughout their lives, because they keep moulting - shedding the whole shell, including the lining of the stomach and the eye stalks, and emerging soft and slightly bigger each time. Old lobsters can be very large indeed; the record is over twenty kilograms.

They also age unusually well. Lobsters produce an enzyme that repairs the ends of their chromosomes, which in most animals shorten with every cell division and act as a countdown. Older lobsters are not noticeably less fertile or less vigorous than young ones, which is why they are sometimes described as biologically immortal.

They are not. The moult is what kills them. Growing a new shell takes more energy the larger the animal gets, and eventually a lobster cannot complete the process - it dies exhausted part-way through, or the old shell fails to come away. What finishes them is not decay but the cost of continuing to grow.