A creature typically can't take one or more free actions during an attack of opportunity because making an attack of opportunity isn't an action
At least, making an attack of opportunity isn't an action as the game defines the term action.
First, let me address free actions. The Player's Handbook presents two slightly contradictory definitions of free actions. Like the question says, Action Types on Free Action does indeed say, "You can perform one or more free actions while taking another action normally" (139). Later, though, Free Actions says, "Free actions don't take any time at all, though your DM may limit the number of free actions you can perform in a turn" (144, emphasis mine, and n.b. a turn is different from a round). In this same section the description of the free action Speak says, "In general, speaking is a free action that you can perform even when it isn’t your turn" (ibid.). No other free action described in that section—like drop an item or drop prone—has a similar special note. This makes the free action speak a largely unique case—and maybe even an abnormal case (see below)—of a free action that can be taken when it's not a creature's turn.
(The above also supports the information provided by this fine answer on how actions are something a creature typically takes on its turn, and the information in this fine answer's succinct quotation from the Rules Compendium.)
Then, on Attacks of Opportunity, the Player's Handbook says
Sometimes, however, a combatant in a melee lets her guard down. In this case, combatants near her can take advantage of her lapse in defense to attack her for free. These free attacks are called attacks of opportunity. (137)
(Emphasis mine.) Thus a creature that's entitled to make an attack of opportunity does not, for instance, take a free action to make that attack of opportunity. Instead, the creature just makes that attack of opportunity, and no action on the creature's part need (or can!) be taken. The free attack that's made as an attack of opportunity is neither a standard, move, full-round, free, swift, nor immediate action; it's not even a not an action action! It's only a free attack, and because this free attack is not an action, a creature cannot simultaneously with it take one or more free actions like a creature can when it takes its actions normally, like it typically does on its turn.
To be clear, the Player's Handbook puts attacks of opportunity before and in a different section from Actions in Combat (138-45) and gives Attacks of Opportunity and Actions in Combat their own unique and independent oversized headers, literally separating the two concepts from each other. Table 8–2: Actions in Combat goes so far as to omit from it altogether an entry for attack of opportunity proper (141).
Further, the Rules Compendium on its chart Actions in Combat puts attacks of opportunity in the section labeled No Action (8), but then clarifies in the chapter Attacks of Opportunity on Making Attacks of Opportunity by saying, "Making an attack of opportunity isn't considered an action…" (18).
Seriously, my heart breaks here. A role-playing game's foremost goal should be explaining—in excruciating detail using multiple examples—who can do what when. Why this game makes who can do what when so incredibly opaque is a mystery.
Long-term implications of this assessment
If this is new information, here're some things to prepare for.
Conditions get messy. More than a few conditions say that a creature suffering from a condition can take no actions, yet the condition doesn't specifically prohibit the creature from making attacks of opportunity (e.g. dazed, stunned). Thus a DM could still allow creatures to make attacks of opportunity while suffering from one or more of those conditions. However, the DM must already deal with numerous issues involving conditions as they are often vague and underwritten. (This contributor has written about the rules-as-written weirdness surrounding the conditions unconscious and fatigued, for example.)
Because of this, this DM approaches conditions with a jaundiced eye and made a house rule saying that A condition that prevents a creature taking actions also prevents a creature from making attacks of opportunity. (That hasn't stopped my players from wanting their PCs to make attacks of opportunity when their PCs are stunned, though!)
The special attack grapple typically can't be completed off-turn. For reasons I don't even pretend to understand, the 3.5 revision added to Starting a Grapple to Step 3: Hold that a creature must "[m]ake an opposed grapple check as a free action" (156) to get the hold, and unlike speaking, for example, there's no special off-turn exception made here. This free action requirement makes it essentially impossible for most creatures to grapple off-turn.
This also means that when Table 8–2: Actions in Combat says in footnote 7 that a grapple can "be used… even as an attack of opportunity," the table, under most circumstances, is largely inaccurate. Although this makes some folks very angry, to this reader, this seems a typical case of text trumps table. (That's the default method Wizards of the Coast's 3.5 errata documents would have readers use to resolve informational conflicts.) (Also see this question for more on grappling and this question for more on source priority.)
Equity for the speechless. If the DM rules that taking a free action to speak is a creature taking an action normally, then creatures that can speak have real mechanical combat advantages over those that can't.
For example, a typical wolf (Monster Manual 283) must take a free action to use its extraordinary ability trip, so during an off-turn attack of opportunity it usually can't use that ability. Nonetheless, a DM that rules that taking a free action to speak is a creature taking its actions normally, then an awakened wolf can make an off-turn attack of opportunity to bite a creature, simultaneously take a free action to speak (rudely—its mouth full of adventurer) to say aloud I bit you! Fall down! then take the free action necessary to use the extraordinary ability trip.
The same would apply to, for example, creatures that are subject to ranged attacks. Those that can take a free action normally off-turn to speak can, alongside that free action, take a free action to drop prone ("Hit the dirt!"). The same luxury would be forbidden to creatures without speech like maybe the common wolf ("Rowr?").1
With the same understanding that creatures that can take a free action to speak off-turn shouldn't have combat options unavailable to their speechless brethren, this DM also has a house rule saying that—because it's taken off-turn—An immediate action is not an action that is taken normally. This prevents creatures that can take immediate actions from having a further combat advantage over those creatures that can't take immediate actions. For example, in this DM's campaigns a creature that takes an immediate action to cast a spell that has a casting time of 1 immediate action cannot simultaneously take a free action to use the feat Quick Draw to draw a weapon.
These are the issues that have arisen most frequently in my campaigns, but a DM may have to address during play a few other edge cases (q.v. this question's answers). And while this may sound like a lot of adjustments to make, the DM already had to house rule many conditions to make sense, this player and DM found 3e's off-turn grappling kind of a pain anyway, and this reader is happier with the leveler playing field that comes from everybody having available similar combat options. In short, these changes aren't nearly as onerous or as destructive as they may seem at first… unless, of course, your the player of a PC whose shtick is off-turn grappling.
1 I hedge here because Dragon #293 includes mist wolves that—and I'm not making this up—"speak the language of wolves" (80). However, this language is never again mentioned in the entirety of the game's corpus (cf. the language Worg (MM 257)).
