Mermaids. Angels. Mister Spock. Speculative fiction sure enjoys a good mash-up, but how likely are we to see genuine human/other hybrids? It depends on what one means by ‘hybrid'.
Agriculture depends on hybridization, of course, but hybrids also occur in the wild. It happens most often between closely related plants, but DNA testing is revealing more animal species that emerged as the result of interspecies reproduction. The Lonicera fly was the first known example of this, but the Pomarine skua is another. Gray wolves and coyotes don't usually play well together in the wild, but sometimes the mood is right and baby coywolves are born. Coywolves aren't considered a separate species, but genetic research suggests that red wolves (extinct in the wild by 1980, now being reintroduced) originally descended from gray wolf/coyote hybrids. If coywolves persist, they may someday develop distinct enough taxonomy and behavior to require a unique latin name.
Unlike wolves and other canids, there are no species on Earth that humans can directly reproduce with. Our closest relatives have a different number of chromosomes than we do, so even if our eggs and sperm were compatible with our cousins' (they aren't), successful fertilization probably wouldn't develop into a viable embryo. Think of the genetic codes involved as sides of a very long zipper. Each parent provides one side of the zipper, but if both sides of the zipper don't have the same number of same-sized teeth, the zipper will jam somewhere along the line. No ‘humanzees', in other words.
However, genes sometimes mingle without reproduction. Sure, microbes are the best at swapping, stealing and installing foreign code, but their own genetic info isn't always the only data that gets transmitted. Transposons, or ‘jumping genes', can replicate themselves and move around a genome. They can cause mutations and disease, and even travel laterally from one genome to another. That's how germs develop resistance to antibiotics to which they are never directly exposed; they share resistant code. Researchers recently discovered that microbes sometimes also deliver transposon DNA from parasitic insect vectors to their mammalian and avian hosts. This naturally occurring horizontal gene transfer between higher animal species might help solve a mystery uncovered when scientists sequenced the human genome: about 40% of our genome is composed of transposable ‘junk' DNA. Does this mean there are human/mosquito hybrids walking among us? Of course not, but given how much they may have influenced human evolution, it might be worth reevaluating our generally low opinion of pests.
We may have quite a lot of genes in common with other animals thanks to distantly shared evolutionary roots and the occasional gene-hacking germ, but interspecies reproduction is simply not an option for humans. There will be no human hybrids, and there will especially not be any human/alien hybrids. (If you haven't already read Athena Andreadis's book, To Seek Out New Life: The Biology of Star Trek, now would be an excellent time to pick up a copy.)
Chimeras are another matter entirely. In nature, chimeras are plants, animals or people with two or more sets of DNA. It usually occurs when one twin absorbs another in utero, but it can happen as the result of a blood transfusion, tissue transplant, and even during pregnancy if a mother's stem cells pass through the placenta and persist in the body of the child (or vice versa). It's an interesting phenomenon in its own right, and then there are the cool and useful man-made mash-ups. Graft-chimeras are fairly common in horticulture, but of course we won't see Daisy-head Mayzie walking down the street any time soon. However, scientists have produced many ‘humanized' animals for medical research, including pigs with human blood, and mice with human brain cells, and other combinations.
It's ethically questionable and highly illegal to engineer chimeras in the other direction – from human stock with non-human grafts. It's true that some people have had pig heart valve transplants, but that's not really the same thing as making transgenic babies from scratch. There's no point rehashing the arguments for and against taking that leap, but purely for the sake of boosting science in fiction, it's important to consider it.
Key to all near-future stories about human graft chimeras is the fact that this is relatively new territory for science. Human testing is problematic on a variety of levels – ethically, culturally, bureaucratically, etc. – but it's further compounded by our long lifespans. Even if a transgenic human is born in a laboratory, it's unreasonable to expect them to spend as much as a century there and never leave. The chaotic world outside a lab wreaks havoc on experimental controls, though, so research will probably stay at the small scale (embryos) for a very long time before doctors can prescribe even specific minor alterations to their patients.
Furthermore, transgenic traits don't have to be inheritable. If only somatic cells are changed, then when chimeras reproduce, their germ cells should produce unaltered offspring. That is, provided the somatic changes don't compromise their fertility or cause their immune systems to reject the embryo's ‘normal' tissues, etc. In other words, mom and dad may photosynthesize, but they wouldn't necessarily make little green babies.
Persistence is also a problem. Remember that a lot of gene expression requires inheritance from both parents. Even assuming that germs cells are also altered, transgenic traits probably won't stabilize and establish themselves unless chimeras are only allowed to reproduce with each other. Without positive selection for those traits, they'll most likely fizzle and disappear into the gene pool within a few generations.
That's actually a good thing! ‘Transgenic' isn't synonymous with ‘perfect'. Just look at corn! There are bound to be unexpected flaws, fatal code errors, and weird consequences (pleitropy). After all, everyone's genes naturally mutate a little over time, and chimeras are already reliably unstable. With the first generations of altered humans, we'll have to learn to expect heavy casualties. Chimeras who survive will probably spend a lot of time in research hospitals receiving personalized, genome-specific treatments invented on the fly for their unique health problems. (Lucky for them, medicine seems to be headed in that direction, anyway.)
Last but not least, there are the actual problems we're up against. Transgenic traits are difficult and expensive to produce and maintain, so they'll probably never be sold as luxury ‘items'. There will be no purely decorative alterations, in other words. But developing human photosynthesis makes sense given how many people die of starvation every year (a number that is set to rise steeply as climate change progresses). And if we want to survive the expansion of our sun and the inevitable destruction of Earth, we'll need to develop adaptations that will allow us to thrive in space and eventually on other worlds.
In the end, there will always be those who reject chimeras as inhuman, abominations and generally unnatural. But the truth is that there are precedents in nature for this sort of gene theft. An amazing sea slug steals algae genes to grant itself photosynthesis, and a weird pink aphid does something similar with fungus and carotenoids. It appears to be a moot point, anyway. According to some scientists, humans are already an artificial ape.