Is it possible for an immune system to exist that makes antibodies but does not possess any kind of immunological memory? If so, what examples are there?
There is no system that makes antibodies but that doesn't have memory.
The title of this question
is was originally very different from the body. The title asks if there's "any immune system without immunological memory", while the body asks if there's "any immune system that makes antibodies but has not immunological memory". That suggests you may think that antibodies are an essential part of an immune system, or that immune systems comprise only antibodies. Neither is true. The vast majority of animals have immune systems that do not have antibodies and that do not have immunological memory; for example, arthropods, the various kinds of worms, etc.
Immunological memory is a relatively recent addition to the immune system, having arisen approximately 400 million years ago, in the lineage that originated with shark-like fish. The species that evolved from sharks -- fish, amphibians, reptiles, birds, mammals -- all have immunological memory, and all have antibodies. Those species that didn't arise from that lineage do not have antibodies, and with two exceptions do not have immunological memory.
The exceptions are lampreys and hagfish, which have a common ancestor with sharks but which ended up forming an immune system that is related to, but often quite different from, that of sharks (and us). Lampreys and hagfish have molecules that are not antibodies but that function in a similar way, including having a form of immunological memory, though it's quite different from ours. So if you're looking for an example of something that kind of has antibodies but that kind of doesn't have immunological memory, that's as close as you're going to get.
- Lamprey Immunity Poised to Evolve
- Harnessing 450 Million Years of Evolution: Lamprey Antibodies for Modern Biomarker Technology
Note that virtually all species, including mammals, have branches of the immune system that don't have immunological memory. Those branches don't have antibodies, though.