Druid/ranger animal companions and paladin special mounts gain hit dice as their master levels, and so gain all the benefits of hit dice, including feats at first and every third level and ability score increases at every fourth level. This did not occur to me until a player asked if this was so, and the answer turned out to be yes, obviously it is so.
The question is fuzzier when it comes to psicrystals and familiars. The consensus on the Rules As Written is that psicrystals do and familiars do not -- but this is naturally absurd, in the same vein as such other strict RAW weirdnesses as drown healing, death not preventing you from taking actions, and tower shields turning themselves (and their wielder) invisible when hidden behind.
Psicrystals and familiars are both treated as having their masters' hit dice and half their masters' hit points, so it should be obvious to anyone with eyes to see that the intention was that they are the same when it comes to hit dice and the benefits thereof. Where the confusion arises is a slight discrepancy in language between the two:
A familiar's hit dice are defined on the sorcerer/wizard entry as "For the purpose of effects related to number of Hit Dice, use the master’s character level or the familiar’s normal HD total, whichever is higher."
The corresponding psion entry makes no mention of a psicrystal's hit dice, so a psicrystal's hit dice are instead defined on the psicrystal creature entry as "Its Hit Dice are equal to its master’s Hit Dice (counting only levels in psion or wilder)".
This is taken by the masses to mean that psicrystals have hit dice, and therefore all the benefits that accrue with hit dice, and familiars are merely treated as having hit dice, and therefore gain no benefits of having hit dice. Despite being baldly preposterous, this interpretation is held by so many people that I feel like I've stepped through a portal into an alternate dimension where nothing makes sense and they spell it "Berenstain Bears" instead of "Berenstein Bears". (I jest.)
Are there any (other) creatures in the game that entirely lack hit dice? I can't think of any. Even animated objects gain hit dice when animated. As far as I know, the game features no creatures that don't have hit dice; everything with hit points and no hit dice is an object.
Hit points are obviously not to be treated as an effect of hit dice here, because HP is defined separately as half the master's HP in the same sections that define familiar and psicrystal HD as equal to the master's HD. Similarly, BAB, saves, and skills are treated separately in separate sections, and are thus probably not treated as effects of hit dice. However, it is perfectly plausible to argue that feats and ability score increases are "effects related to number of Hit Dice", and thus accrue to familiars even if you somehow maintain that familiars really don't actually have hit dice.
Moreover, the line under Familiar Basics that qualifies "For each skill in which either the master or the familiar has ranks" clearly indicates that familiars have skill ranks, which they couldn't have if they don't have hit dice. It goes on to say "use either the normal skill ranks for an animal of that type or the master’s skill ranks, whichever are better", which seems to indicate that a familiar doesn't gain skill ranks beyond what they start with, but it clearly still has them, which means a familiar must have at least its starting hit dice.
In any event, the Rules As Written and the Rules as Intended should always be subservient to what makes sense and what makes fun. The RAW, being vague on what an "effect related to number of Hit Dice" is, are not as clear as everybody thinks they are, the RAI are clearly in conflict with the common interpretation, and, though what makes more fun is unclear (casters don't really need nice things; much of the familiar section seems intended to simplify the familiar for ease of play but actually has quite the opposite effect), what makes sense is that the things that are the same should be the same. Either both psicrystals and familiars have feats, or neither does.
This steadfast adherence among the GITP forums (at least, every member of the GITP forums I've talked to about it) to this particular interpretation of RAW, in defiance of other interpretations of RAW and all the considerations that should be more important than any interpretation of RAW, is a prime example of their fetishization of the consensus interpretation of RAW, however nonsensical, above all the considerations that should be more important in actual play. (Which is why I'm posting this here, in this space safe from RAW fetishization, and not there.) Don't get me wrong, the GITP forums are a great place full of great people, but they can get pretty dogmatic about their assumptions sometimes.