In part the answer to your question is what you mean be an animal.

There are bacteria (which are not animals in anyone's book) that can reproduce once every 20 minutes or so.

Some unicellular yeasts, which are eukaryotes, can divide once every hour or so.

Based on my own survey of the literature, the fastest reproducing metazoan (multicellular animal) may well one of the salps. Salps are members of the phylum Chordata (the same phylum that we are in!).

Salps are tubular, gelatinous animals that live in the open ocean and feed on plankton -- they eat and move by jet propulsion, pumping water through their bodies.

They alternate between a sexual and an asexual phase. The asexual organisms live as isolated individuals and reproduce by budding.

A new organism can be formed in less than a day.