In real life, it's impossible for humans to breed with non-human animals due to the difference in DNA. The realms of fantasy and science fiction are less strict about the possibilities of human hybrids, however, with some of the most popular characters in fiction being half-humans, such as Spock from Star Trek.
Dungeons & Dragons has several monster races that can breed with humans in order to make new hybrid races, some of which don't make a great deal of sense. It seems that no one ever taught the creators of Dungeons & Dragons about the birds and the bees, as some of these couplings would not exist if they did.
Elves & Orcs
Half-elves and half-orcs have existed since the first Player's Handbook of Advanced Dungeons & Dragons. The concept of half-elves and half-orcs was popularized by The Hobbit and The Lord of the Rings, with characters like Elrond being a major part of the story for the former and the creations of Saruman making up the latter. The early editions of Dungeons & Dragons borrowed heavily from The Lord of the Rings (to the point where a lawsuit ensued), so it's easy to see why half-elves and half-orcs are part of the game.
Dragons might be giant lizards, but they are some of the most fertile creatures in Dungeons & Dragons. Humans are not the only species who can give birth to half-dragons in Dungeons & Dragons, which is due to the magical powers that most dragons possess.
The way that dragons breed with other creatures is through shapeshifting. As a dragon grows older, it will gain the ability to cast spells as if it were a sorcerer. Once a dragon acquires the ability to cast the polymorph spell, it can take on another form and impregnate (or be impregnated by) another creature, leading to the birth of a half-dragon.
The power within a dragon's blood can give the humanoid races the ability to cast magic and many sorcerers owe their powers to a dragon relative in their family tree.
The angels of the Dungeons & Dragons multiverse possess the ability to crossbreed with mortals, which is a result of their magical nature. There are many celestials that resemble winged humanoids (like the solar), so the idea of them breeding with humans and other humanoid races isn't so far-fetched.
Half-celestials are incredibly powerful beings, with the ability to cast a select few cleric spells for free every day. The descendants of celestials are commonly part of the aasimar race and the power of the angels in their blood is diluted, but that weakness often makes them more suitable to be player characters than half-celestials are.
Demons & Devils
It isn't just the denizens of the Seven Heavens who are getting down with the mortals, as the inhabitants of the Lower planes are also available for inter-dimensional booty calls.
The fact that there are demons and devils associated with indecent acts makes the idea of half-fiends existing more understandable than most of the hybrids on this list. The incubus and succubus exist for the very purpose of tempting mortals and they seemingly have no fear of being dragged onto The Maury Povich Show.
The tiefling race is composed of those with fiendish blood in their veins. It seems that the Necronomicon must be the new alternative to Tinder in the Forgotten Realms, as tieflings are now in the Player's Handbook in the current edition of Dungeons & Dragons, which means that a lot of them must be running around.
A Tornado With Angry Eyes & A Pool Of Water
The third edition Manual of the Planes lists the half-elemental template which can be added to different mortal beings. It says in the book that it's possible for a half-elemental to be born from a union between a mortal being and an elemental.
An air elemental is essentially a tornado with a face and a water elemental is a big pool of water.
Are the elementals into it? Do they have to be coerced into performing? Is there an elemental equivalent to furries? Do they go conventions dressed like their "humansona" and wander around in groups?
The Thing & An Inferno
The next two elementals in the dating game are the earth elemental and the fire elemental. The earth elemental is a creature made entirely of stone, while the fire elemental is a walking column of flame.
It seems that the elementals are considered to be far more attractive than we are giving them credit for, as there are several half-elemental members of the cult of Tharizdun in Oerth. There are also elemental equivalents to the aasimar/tieflings, known as the genasi, who started cropping up in Faerun during the third edition era.
Aladdin used one of his wishes for the sole purpose of trying to impress Princess Jasmine. It never occurred to him that his true love might have been the annoying blue guy in the lamp the whole time.
The weakest of the genie race are known as janni and they have to spend most of their time on the prime material plane, which means that they form societies in the desert and often encounter humans. There are quite a few half-janni running around in the hottest places of the Dungeons & Dragons campaign worlds.
The idea of there being human/vampire hybrids is a popular one, with series like Twilight, Blade, and Vampire: The Masquerade using the concept of dhampir to prove that even the dead are capable of knocking someone up.
The book called Libris Mortis established that a vampire who had feasted on enough blood would briefly restore enough of their biological functions to sire a child. It's also possible for demihuman/humanoid vampires to perform this feat. The half-vampire children possess some of the powers of their undead parent (including the ability to charm mortals and resist damage from non-silver weapons), but they have to feast on blood in order to function properly.
The fey have spent a lot of time interacting with mortals, as they love to use their magic to beguile those who tread upon their domain.
There are many fey who are similar in appearance to elves and many of them possess the magical ability to alter their form, so it's no surprise that many half-fey creatures have been born into the world. Those with fairie blood are strong candidates for becoming sorcerers, which means that if your character's grandmother had a fling with Jareth from Labyrinth, then there is a good chance that you can become an arcane spellcaster.
There are some people who love their kids so much that they would be willing to get down and dirty with a hideous green monster in order to give their children some awesome regeneration abilities.
The trolls of Dungeons & Dragons are monstrous green giants who exist only to devour and destroy. Trolls also have a powerful regeneration ability that allows them to survive almost any wound that doesn't involve acid or fire, which means that some people are willing to try and create half-troll children. The Fiend Folio specifies that some spellcasters are interested in creating half-trolls in captivity in order to make beings with the regeneration and brute strength of trolls, but that were easier to control.
Denizens Of The Far Realm
The book called Lords of Madness established that there are some incredibly freaky mortals in the Dungeons & Dragons universe, as there are people who seek out Lovecraftian horrors for romantic dates.
The half-farspawn is created by a union between a mortal being and a denizen of the Far Realm. The Far Realm is a plane that is home to many of the aberrations that plague the Dungeons & Dragons multiverse, which include the likes of aboleths, beholders, carrion crawlers, and mind flayers. These are all creatures known for consisting mostly of tentacles (or eye stalks), which suggests that the existence of Japan wasn't necessary to cater to certain kinds of fetishes in places like the Forgotten Realms.