RAW: Once and for all.
Unfortunately, the Knowledge skill never got much love, despite how crucial information is to the game.
The rules are simple, simplistic really, and as you noted do not allow re-rolls. Ever.
This leads to 2 unfortunate effects:
- Each character should keep a complete list of all the Knowledge checks they did and their result, which is a lot of book-keeping.
- A character which does not know something at level 1 will still be ignorant of it at level 20, even if they spent all their free time trying to learn about it.
I would encourage any game to alter the Try Again entry. It's unlikely you'll come up with anything worse.
My proposal: once per encounter, with additional rolls whenever additional information flows in.
Rather than treating a Knowledge check as what a character knows, I encourage treating a Knowledge check as what a character remembers right now. This helps with the verisimilitude, as it explains why a character could remember something once, but not later.
Mechanically, with regard to identifying a creature, I advise starting with the rules of the Knowledge Devotion feat (Complete Champion, p. 60):
Whenever you fight a creature, you can make a Knowledge check based on its type, as described on page 78 of the Player's Handbook, provided that you have at least one rank in the appropriate Knowledge skill.
You can make only one Knowledge check per creature type per combat. If you fight creatures of multiple types during the same combat, you can make one Knowledge check per type, thereby possibly gaining different bonuses against different opponents.
With a slight adaptation for forgetful characters:
- Per-encounter tracking: to avoid lugging around an unwieldy list of everything each character knows or doesn't know.
- Once for free, whenever a creature of this kind1 first appears.
- Once each time the creature exhibits a new ability2.
I would also remind the Dungeon Master than they should feel free to give Circumstance bonuses at their convenience or to simply reveal information outright.
If the party already encountered a Formian Taskmaster an hour ago and it Dominated a party member, then upon recognizing another Formian Taskmaster the character should obviously remember that it may be able to Dominate a party member3.
Similarly, while a Knowledge (Religion) check covers all kinds of Undead, a character who has specifically spent time researching Mummies should be afforded a +5 or +10 Circumstance bonus4 on a check to identify Mummies.
1 It's unclear whether the Knowledge Devotion feat really intends for a single check to be rolled for all Humanoids, regardless of whether they are Gnomes, Elves, Humans or Halflings or for all Outsiders (oO). I advise reading type not as the Type (Humanoid, Giant, ...) but instead as the Kind: for example, one roll each for a Formian Worker, a Formian Warrior and the Formian Taskmaster overseeing them.
2 Unlikely to be game-breaking, and helps players feel better about their investment in Knowledge skills. For this purpose, ability means any Natural, Extraordinary, Supernatural or Spell-like Ability, which tend to define a creature, but not feats or skills which vary from individual to individual.
3 The DM should start by reminding the character of the abilities it already knows about the creature, either those it recalled or those it already witnessed. To avoid metagaming, the DM may ask the players what they know of the creature, and offer those abilities/facts first; even the best players may otherwise be tempted to act on information they know but their characters doesn't, such as fighting a Medusa blindfolded.
4 For each increment of +5 a check beats the DC, the character remembers one more ability or fact about the creature; thus it makes sense to use +5 bonus increments on the check for rewarding familiarity.