In the Basic editions of D&D where elves were a class rather than a race, Interspecies Romance between a human and an elf typically resulted in either a human or an elf, much like in Tolkien's world.
Eberron gives a very reasonable explanation to why half-elves are a race, and lays down specific rules for half-elves and their offspring (half-elves breed true, elves and half-elves always have half-elf children, and a half-elf/human pairing has an equal chance of producing a half-elf or human). When the elves first started trading with humans, they realized that humans were very short-lived and could die in a matter of decades. Some elves decided it would be good business to marry wealthy human merchants, enjoy the relationship for a few decades, and then inherit sizeable holdings. It didn't even occur to the elves that half-human/half-elf offspring were viable, and when half-human children started being born the elvish nation restricted trade and closed its borders out of fear. However, all of the newly born half-elves were born into rich families, and most of the elven parents stayed around to raise their children, so half-elves came to occupy a nice section of upper/middle-class society in Khorvaire and make "Khorvar" villages on their ancestral land holdings.
In fact, orcs likely have the most unstable genetic structure of all humanoid beings. One source claims that can crossbreed with almost any humanoid race except for elves, and have done so with goblins, dwarves, and gnomes, as well as humans. Two well-known half-orc species that don't involve a human parent are orogs (which is the result of male orc and a female ogre, and somehow smarter than its parents and much more disciplined than either) and an ogrillion (a rarer creature that occurs with the same two species, but the genders of the parents reversed; more stupid than both parents and, somehow, armored).
Maybe, but remember half-orc/half-elf are still taboo (strange thing since orc and elf in D&D are from Tolkien, who described orcs as tainted elves...).
Derro and all Gith subraces are results of slave breeding by illithids.
4th edition D&D also changes the Derro to being a culture of aberration-worshippers who have devolved into mad, dwarf-like creatures on their own.
D&D3+ also features, as templates, half-celestials, half-dragons, half-elementals, half-fey, half-fiends, half-janni, half-minotaurs, half-ogres, half-trolls (which can be anything from half-human to half-griffin to half-stegosaurus), half-vampires, and even half-golems and half-illithids (though at least those last ones thankfully don't involve sex).
Thankfully?! The process that can produce a half-illithid is worse than rape.
"'Half-dragon' is an inherited template that can be added to any living, corporeal creature." Oozes are "living, corporeal creatures", which makes the Half-Black-Dragon Gelatinous Cube a popular joke monster.
Going further still, aasimar (celestial), chaonds (chaos), draconics, genasi (elemental), tieflings (fiendish), and zenythri (law), among others, are what happen when some of those half-thingies go around and do it with "normal" people.
There are also "Heritage" feats and a "Bloodline" option, for adding special powers based on unusual ancestry, including celestial, fiendish, fey, vampire and illithid, among too many others to list.
In fact, it was implied that humanoid sorcerers have some draconic ancestry, as sorcery is how the dragons use magic - and they're pretty much the oldest race, outside of celestials and fiends - as well as explicitly stated that dragons can mate with Anything That Moves. Which fits well with their high Charisma and shapeshifting abilities. Nowadays it's implied that sorcerers have a touch of magical blood, but this can be from dragons, angels or demons, with different effects depending on which.
D&D was so bad about everything mating with everything else that the infamous third-party sourcebook, Book of Erotic Fantasy, featured a chart explaining which creatures are compatible with which other creatures. It did have some interesting things in it, like prostitute prestige classes and cloud giants mating with sprites.
Dragonlance is not immune from this trope: the Dragonlance setting features 'Gully Dwarves', allegedly the offspring of gnomes and dwarves. Aside from being a strange combination, Gully Dwarves are incredibly stupid creatures, depicted as being totally incapable of counting higher than two. Those that can count higher than two tend to lick beer from tavern floors.
The changeling race, from the Eberron campaign setting of Dungeons & Dragons, are theorized to be descended from the viable offspring of humans and dopplegangers coupling, though this is unconfirmed (various factions have their own varying theories about why, exactly, changelings exist).
Note that in Standard D&D this is explicitly not the case: Dopplegangers are a One-Gender Race that breed with other races, their children turn into Dopplegangers at puberty.
The game plays with the idea of satyrs chasing after nymphs; it does happen, yes, but one source claims that might not qualify for the Trope at all, saying that the two races might be male and female counterparts of the same species. (Satyrs are Always Male, while nymphs are Always Female, and for either species to procreate, they have to do so with each other; if a nymph conceives a child from such a union, a son is a satyr, while a daughter is a nymph. If either species mates with a human, the offspring is a Half-Fey.
Shifters or "beastkin" are similarly referred to as being a "mixed race" of lycanthropes and humans, although this, too, is unconfirmed and fiercely denied by shifters who belong to the anti-lycanthropic Church of the Silver Flame. (Such shifters insist that they existed first and lycanthropes are an unholy union of humans and shifters.)
The Fourth Edition Monster Manual and Player's Handbook 2 do make this the default canon explanation how the shifter race(s) came to be; longtooth shifters are descended from werewolves, razorclaw shifters from weretigers. Both are legitimate choices for player characters.
Forgotten Realms even has specific breeds of tieflings based on their ancestry - specifically, the fey'ri (elf-demon crosses) and tanarrukks (orc-demon crosses).
Acording to Dwarves Deep, humans, gnomes and halflings are cross-fertile with dwarves. And it's not just a theory, but accepted practice among the Shield Dwarves: dwarves' fertility is dangerously low (due to exposure to nonorganic poisons, later The Spawned taint was added) and the quarter-bloods will be proper dwarves.
Dwarf-elf pairs did exist too, but mostly in Ye Goode Olde Days of Ardeep and Earlann, or at least Myth Drannor. Which was named so after Drannor Whitethistle, who married the dwarf lady Konora Onyxhelm. Their kid Labrad became one of the first settlers in founding Cormanthor and apparently combined orderly hard-working side with wood-loving one, considering he's known as "the First Gardener". Another dwelf is known as a runecarver and archmage.
The tendency for humans to mate with other things was so common that eventually a race was created in D&D called the "mongrelfolk," supposedly a lowest-common-denominator mish-mash of basically *all* humanoid races.
Then there is the tauric template. Not happy with centaurs? Pick a humanoid and a creature with four or more legs and mash 'em up. Just try not to think about how they came to be. (Thankfully, the answer is usually "magic".)
Mechanatrixes, from humans and extraplanar clockwork outsiders like Inevitables. Born as cyborgs. From a living being and a magical robot.
Although the Squickiest race is quite probably the Wildren, a race in the 3.5 edition Planar Handbook that are descended from crossbreeds of dwarves aaaaannnd (wait for it)... badgers. Yes, badgers. Admittedly, they were near-sentient celestial badgers that had quite possibly previously been dwarves because of the way that their native plane works, but still. You may find that the best way to gouge out your mind's eye is to ram a tuning fork up your nostrils.
They've got some competition from the Aellar; elf/giant eagle crossbreeds, appearing as wiry elves with wings, brought into being by elf druids who studied for years in order to learn how to change into giant eagles... then promptly used that power for nookie.
Elf/dragon hybrids, although perfectly feasible (with shapechanging assistance) under 3E rules, are such an unthinkable taboo to both species in the Eberron setting that producing one got the Death-dragonmarked elven lineage of Vol exterminated. The hybrid still exists, but only as a lich.
For non Planescapeers, hybrids of Red Dragon and Githyanki, deliberately bred by the Lich-Queen Vlaakith.
According to Dragon Magazine #385, human royals have been known to accept Glee-Born, dragonborn with a more jovial, fun-loving nature than is normal for their taciturn race, as courtesans. Dragonborn, by the by, are humanoid in form, but have four fingers and three toes to a hand/foot (all of which are tipped with big, but blunt, claws), have scales for skin, "dreadlocks" for hair (actually a specialized form of scale), blunt-muzzled reptilian heads, fin-like ears, and grow close to seven feet tall. And typically have physically proportions like dwarves... meaning that some dragonborn women are bigger and buffer than most human men. Of course, there are still relatively slender and feminine dragonborn, but still.
Al-Qadim, being quasi-Arabic setting, has polygamy limited by tradition to four wives. That being Al-Qadim, this number is justified by the legend about the first sha'ir, who had four genie wives — one per element. Marriages between a mortal and a genie are not quite stuff of legends, though of course rare and many are troublesome — given how capricious and powerful genies are, few can hope to hold one's interest for long, let alone be a more or less equal match. Mortals of Zakhara are immune to Fantastic Racism, so romance between humanoids happens now and again, though rare except human-elf and human-orc pairs — they are known to be inter-fertile, so it's much less of a big deal.
While it doesn't get a lot of mention, the novels for the Dragonlance setting admit outright that humans can cross-breed with just about any of the humanoid races. Not only is a half-elf one of the main characters of the first trilogy, but several mentions have been made of half-kender, and at least one half-ogre, half-dwarf, half-goblin and half-gnome has each turned up over the course of the novels.
Consequently, the tabletop game has a lot of half-race writeups, particularly in 3rd edition sourcebooks.
This troper heard somewhere that originally, Gnolls were gnome-troll hybrids rather than anthropomorphic hyenas.
While a third-party sourcebook, "Bastards and Bloodlines" is highly illuminative of how much use Dungeons & Dragons can use this trope. In addition to the above mentioned Aellar and Half-Ogres (here presented as both human/ogre and orc/ogre crosses), a large number of races only born through magic, it includes:
Alicorns, a hybrid of elf and unicorn, typically born when an elven ranger or druid grows powerful enough that they attract a unicorn of the opposite sex as an animal companion. These have a "Dark Alicorn" counterpart, born of black unicorn/drow elf pairings.
Decataur, an elf/centaur cross that is actually either a biped or a quadraped depending on whether the centaur was the father or the mother. They are one of the comparatively few explicitly sterile races in the book.
Green Folk, a human/lizardfolk crossbreed that is sometimes created through sex, but more commonly a result of failed resurrection magic on a human near lizardfolk territory. Males are sterile, females can breed with either "parent" species but only have sterile male offspring.
Grendle, the result of crossbreeding trolls with either dwarves or humans.
Half-gnolls, born of similar situations as the better-known half-orcs.
Half-goblinoids, the sterile results of goblin/gnome, goblin/halfling, dwarf/hobgoblin and elf/bugbear breeding.
Half-kuo-toa, a rare breed of hybrid spawned from the union of kuo-toa and coastal dwarves.
Houri, the daughters of nymphs who take male elves as their lovers.
Jovians, the sterile offspring of hill giants, fire giants and frost giants with humans or orcs.
Kestrels, the hybrid offspring of harpies and halflings.
Mergs are the result when humans and merfolk have children.
Mind Rippers are half humanoid, half mind-flayer, and strongly implied to be born... the traditional way.
Morlocks, ghastly gnome/troglodyte crossbreeds.
Pipers can result from elven or halfling women who dally with satyrs.
Spring Children are the incredibly rare results of dalliances between dryads and humanoids.
Stheins are the result of breeding elves and nagas of any combination.
Trixies are gnome/pixie hybrids brought about by relationships between the two species.
Wendigoes are born when dwarves driven to feral madness couple with winter wolves after running with the pack for a sufficient amount of time, though they are fortunately sterile.
Woodwose are produced when an treant falls in love with an elf who comes to reciprocate its feelings.
Wretches are born when a hag chooses to take an orc as her mate.
Wyrds are descended from unions between elves and ogre magi, often the result of bargains struck between the ogres and drow.
A rather twisted, Mad Love version can be seen in the Book of Vile Darkness for edition 3.5, involving a masochistic, self-mutilating female medusa Cancer Mage named Siddal, and a male half-orc Vermin Lord named Gauderis. He is in love with her, though she isn't aware of his feelings, and his greatest wish is to be petrified by her gaze, something that he believes will be an ecstatic experience. The Body Horror of each just makes this relationship even more sick and twisted.
The 3.5 edition module Return to the Temple of Elemental Evil introduces half-elementals, incredibly rare creatures that are the result of a union between mortals and elemental beings. The mortal is not necessarily human; according to the template, it can be "any corporeal creature" that has an Intelligence score of 4 or more. This can include some undead creatures, by the way, which normally can't sire or bear children via regular means. Of course, because elementals are usually not even anatomically compatible with mortals, the book mentions that magic of some sort is usually involved in their conception. The ones who appear in the module are all villains who worship Tharizdun, who allied himself with many powerful evil elementals in his guise of the Elder Elemental Eye. (One of the three most powerful leaders of his cult is a half-elemental.)