I've been inspired to ask this question based off of my answer, and the resulting discussion, to this question: Why would a species of intelligent parthenogenetics invent males? However, this is actually something that I had contemplated many years ago. A species I had thought was an interesting idea, but wasn't certain how it would work.
The idea was for a creature, not necessarily sapient, that evolved a way to 'fast-forward' evolution by 'recognizing' that mutation, despite the risk of their being maladaptive, can occasionally lead to beneficial adaptations; and thus evolved a way to allow/encourage additional mutations in at least some of their young. These mutated young would sometimes develop positive mutations could be spread to the rest of the species, thus encouraging development of new adaptations faster than would be expected compared to similar species of their world. (yes, I'm personifying evolution here to make it easier to explain the idea. it's sometimes easier to say 'fast-forward' then to discuss actual specifics, I think the intent of my shorthand is clear if you presume I understand how evolution really works).
I'll go into the ideas which I toyed with when younger for this species, but I'm not committed to them. All I'm interested in is an evolutionarily viable species which has evolved a way to encourage adaptation through genetic mutation (on the long run, evolutionary changes are still going to be on geological time scales). Any extrapolation on what this would mean for such a species is interesting, but the important thing is how it would evolve, and how the adaptation would work (and be worth the tradeoff of increased birth defects etc); perhaps how it would affect their mating strategies.
I'm interested in a large, complex species; say at least as 'complex' as a wolf, not bacteria or simple worms.
The species must 'choose' to encourage mutation in some way; not have it forced on them by their environment. Just having a normal species in a world with extreme mutagens is not enough. Instead this should be a controlled adaptation. This means either the species has some way of controlling how much/when to allow mutation of young to occur, or some means of causing mutations to occur in a way that they do not suffer from the increased triggered mutations.
Below was my original idea, which I don't think quite works looking back on it now, with more experience. I include it to give a possible idea of tricks that can be used, perhaps someone would want to work with this species idea and just adapt it to be more plausible. However, I'm more then happy to hear about entirely different species with controlled mutation.
My species lived on a world that had some form of extreme mutagens, perhaps one where their embryos were particularly prone to mutation. The species has evolved a way to protect the embryos from mutagens when they were developing, perhaps by placing the uterus next to a thick carapace that did a good job of deflecting the mutagens while the young were in their most vulnerable embryonic state. This way they could safely carry 2-3 young to term with no more risk of mutation then we have.
However, the species also had a second unprotected womb, perhaps part of the same womb that did not 'fit' under the carapace for protection, or else she could distribute weight more evenly by carrying young in two wombs at different parts of her body, but only one could be protected with a carapace. The female could choose to mate and conceive using either womb, but would prefer the protected womb to avoid birth defects. However, in times of plenty when she is capable of tending to more young she may choose to mate again and carry an extra young in the unprotected womb, one that is not protected from the mutagens. This young would be more prone to birth defects from mutation though, and thus would usually be the 'runt' of the litter; only bothered with if she was certain she could dedicate resources to raising the runt without limiting her ability to care for her current litter. He likely would always get the last priory of resources, she may even spontaneously abort him if she realizes she has less plentiful resources later.
However, some mothers will choose to have an extra runt, and some of these runts will be born health, survive and mate. These runts, with their much higher mutation rate, would also be more likely to develop adaptations, and thus the runts are the most common driver of 'evolution' by bringing in new novel and potentially beneficial mutations to the gene pool.
I had liked the idea at the time, as it could even allow me to develop some interesting psychology for the species and how they would treat the runts. However, I'm not certain I buy the two-womb idea now. The idea was that the runt was that they couldn't protect any more young from mutation, so they would settle for a runt despite his mutation risk when they had excess resources (because if you have the resources, why not take a gamble of an extra young), and not bother with one if they can't afford to raise another. However, evolving a whole second womb seems more complex than just developing a way to protect the runt from the mutagens they way the others are. Why support an extra womb if it's not often used, and why not figure out a way to protect that womb as well etc.
Perhaps my original idea can be adapted to work, perhaps not. The only challenge I have is to create a believable species that has some form of encouraging mutation in their young. I look forward to seeing what others come up with.