(With help from Peter Lansdorp of the BC Cancer Agency)
The thing you have to understand is that the telomeres are supposed to be "recharged" or reset each reproductive cycle. This is thought to occur in the germline (in the gonad). The animals cloned by humans do not pass through this stage and hence do not get reset. This can also occur in early embryogenesis.
Many asexual animals reproduce via a germline mechanism (oocytes remain diploid and then develop for example), these of course get reset.
The other types of asexual reproduction present a real problem. How many generations can you cut a starfish in half and still have it reproduce? I do not know.
I suspect the animals that routinely use an asexual route have a reset method, but I do not know if it has been investigated.... A good research project!