What organism has the longest Hayflick Limit?
The Hayflick limit is generally associated with telomere length. Human telomeres are a little on the long side as species go, but are not extraordinary. Many species of mice, and other rodents, have far longer telomeres than humans, for example, and obviously have much shorter lifespans. There's also such a thing as a "mega-telomere", found in a number of bird species, which can hundreds of times longer than human telomeres.

There's only a vague correlation between telomere length and a species' lifespan.  See Comparative biology of mammalian telomeres: hypotheses on ancestral states and the roles of telomeres in longevity determination for a starting point.  Telomeres are a part of the longevity story, but probably a fairly small part.

That said I don't see an explicit statement that those species with very long telomeres have a higher Hayflick limit. I assume this is because the Hayflick limit is an ultra-simplistic and not very useful description of a very specific set of conditions, but I don't know that for sure.