In my campaign(s), several characters have pets, some have had children, some have special mounts. Here I'll talk about some of the above, and try to give appropriate credit to the sources from whence they came.
In a current campaign set in the world of Greyhawk, the group happened across an NPC with a Jhereg. For conventional AD&D reference, use the stats and likeness of the Pseudo-dragon in the Monster Manual. For more insight into the creature and how it interacts with it's human, read any of the Jhereg series from Steven Brust.
From the point where first contact with this NPC was made, the members of the group decided that they had to have one of their very own. It took some investigation on their part, as well as going a good distance out of their way, to locate the hatching grounds of the Jhereg. Now two of the three have Jheregs, both bonded in the presence of the creature's mothers. The creatures can communicate telepathically with their humans, and can carry out scouting tasks from the air for them. They are also able to detect invisible, and carry a non-lethal poisonous sting in their tail. Both owners are female, and if their Jheregs ever die, I know I'm going to have tears to deal with...
The third member of the group has a Cooshee. Refer to the Monster Manual II for information on this puppy. This creature was obtained from an elf who was selling them at an outdoor market outside of Highport. He only sold to elves and half elves, and then only if one of the litter took a liking to them. Since obtaining this elven canine (who weighs more than she does), they have become good friends. In one adventure, a being of great power endowed the Cooshee with the ability to communicate mentally with her owner. For some reason we don't need a third Jhereg anymore...
Several times characters have purchased War Dogs in kennels and marketplaces. The problem is, a dog bred for combat is easily provoked into it. None of the War Dogs purchased so far have had much luck hanging around for long.
One of my favorite mounts doesn't have a conventional reference. It's one of those things from my own sick imagination. They resemble horses, but have clawed feet, sharp teeth, powerful jaws, and crappy attitudes. These beasts are intelligent, and usually evil. With some magical abilities which they only manifest when they feel like it, these creatures maintain an aura of mystery and mistrust. One unfortunate trait of these critters in their love for horseflesh. They will attack and eat defenseless horses, say, locked in an adjacent stall. Some can communicate telepathically with the one they choose to allow to ride them. If you have one in a party, keep the other horses clear of him, as well as anyone who may assert their dislike of the mount. If this nasty creature appeals to your more basic greedy instincts, you may call it a Kurram.
Another creature from the same cluttered realm is the Dragasus. Having the general shape of a Pegasus, with horns that lie back along the top of the skull like flattened ears and great bat-like wings, this creature is usually considered an encounter rather than a treasure. It does have something of a breath weapon, though it's fairly weak, doing only 2-8 points of fire damage. As mounts they move and function as a Pegasus, though they aren't concerned with the alignment of the rider, so much as the treatment they receive. Most Dragasus are red in color, their fur somewhat longer than that of a Pegasus. By the way, they have a racial hatred for Kurram.
Both of the above creatures, while useful and entertaining as mounts, may cause trouble when brought into towns and cities. Have a care when stabling either a mount that attacks horses, or one that could cause a stable to burst into flames if it sneezes. You get the idea...
The current most popular mount (aside from dragons, of course), is my Destrier. While I realize that there are references to this mount available, I contrived my own version for my campaign. These stand over 25 hands high, have two eyes on either side of their heads, and can move at a reasonable pace (12-20") for several days at a time. These monsterous mounts do resemble draft horses, and getting into the saddle can be very arduous, especially for smaller demi-human races. As two of the Destrier's owners are half elven, this was eventually overcome. Initially they had rope ladders to mount, and had to have the humans in the group do the saddling and unsaddling. In my Greyhawk, only elven cities have the resources and know-how to raise these creatures.
While some adventurers would consider a child as a consequence rather than a boone, there are some semi-retired characters who have had children, some by NPC mates, some by other characters. In most cases when this happens, the mother (usually) is sent out to pasture, to be replaced by another character. (Sir Beef's kids have a castle staff to take care of them, so he and the wife can sneak away and go adventuring occasionally.)
One child that is radically different in one of my longer running campaigns is the kitten of a Hengyokai character. (See Oriental Adventurers, character races.)
The mother is a Hengyokai cat, and the father was similar enough to effect this offspring. Fortunately the mother is high level and well established in her city, employing household staff to help take care of the child. While the mother is still played and enjoyed by the player, one day there may be a legate for the player to begin as a new character.
I subscribe to the suggestion that legacies can be very dangerous and unfair within a campaign. Two characters who are obstensibly related, should not be able to share wealth and magic items with one another. If the previously mentioned offspring ever developes to a playable age, some method will certainly be used to avoid starting it out with Fort Knox.
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