Greatsword, headbutt, and bite during one full attack? Yes…
A barbarian 6/fighter 6 that's wearing medium or heavy armor that makes a full attack can make his manufactured weapon attacks normally then, because he's made a full attack, also use the extraordinary ability steel headbutt using his base attack bonus but at −5.
During that same full attack, such a warrior can employ any natural weapons he possesses—such as the bite granted by the rage power animal fury—as secondary natural attacks (therefore making each natural attack as if his base attack bonus were −5 lower and applying only half the warrior's Strength bonus on the the natural attack's damage roll).
A GM may limit a warrior's use of some natural attacks depending on the manufactured weapons he's employing during the full attack. For example, most GMs—including this one—rule it's impossible to use during the same full attack both a two-handed weapon and claws as the GM and the game often prejudiciously assume claws to be on hands instead of feet. Likewise, as Gnomejon mentions in this answer, a GM may rule a creature can't make a headbutt and a bite in the same round. (This GM, however, would allow a headbutt and a bite in the same round. It's not like a creature is doing both simultaneously. That is, a creature capable of multiple attacks with its greatsword isn't somehow making those attacks all at once. Combat is an abstraction, a turn representing an exchange of blows as well as parries and dodges and so on—not individual attacks. Were turns representative of actual events, everybody'd call it Slothfinder.) (For comparison, in GURPS events are actual instead of representative and turns are 1 second long.)
Anyway, more about natural attacks in Pathfinder can be learned from this question, this question, this question, this question, and more than a few others, too. (Apparently, Pathfinder should make this topic a little clearer.)
…But be careful with those attack bonuses
When a creature uses its base attack bonus −5 for something like a steel headbutt or secondary natural attack, it does not just get to use its highest attack bonus −5. Instead, the warrior subtracts 5 from his base attack bonus and computes his attack bonuses for those attacks separately and individually, adding bonuses and suffering penalties that apply only to those attack forms. For example, the enhancement bonus that applies to attack rolls with the +3 greatsword because of that weapon's +3 enhancement bonus don't count toward the attack bonuses of the steel headbutt and the bite.
(In other words, greatsword +3 at +23/+18/+13 and steel headbutt at +15 and bite at +15 is correct.)
A Medium creature normally can't wield a Large 2-handed weapon
The titan fighter extraordinary ability giant weapon wielder allows, for example, a Medium warrior to wield a Large two-handed weapon in two hands. Weapons on Weapon Size says that this is usually impossible:
In general, a light weapon is an object two size categories smaller than the wielder, a one-handed weapon is an object one size category smaller than the wielder, and a two-handed weapon is an object of the same size category as the wielder.
The measure of how much effort it takes to use a weapon (whether the weapon is designated as a light, one-handed, or two-handed weapon for a particular wielder) is altered by one step for each size category of difference between the wielder's size and the size of the creature for which the weapon was designed. For example, a Small creature would wield a Medium one-handed weapon as a two-handed weapon. If a weapon's designation would be changed to something other than light, one-handed, or two-handed by this alteration, the creature can't wield the weapon at all.
Inappropriately Sized Weapons: A creature can't make optimum use of a weapon that isn't properly sized for it. A cumulative –2 penalty applies on attack rolls for each size category of difference between the size of its intended wielder and the size of its actual wielder. If the creature isn't proficient with the weapon, a –4 nonproficiency penalty also applies.
Emphasis mine. That means a typical Medium creature that attempts to wield a Large two-handed weapon must wield that weapon as something other than a two-handed weapon, and, therefore, the Medium creature just can't wield it. However, the extraordinary ability giant weapon wielder allows a Medium warrior to employ a two-handed Large weapon but the warrior suffers an additional –2 penalty on attack rolls with it, for a total of a –4 penalty on attack rolls (the other –2 penalty coming from the weapon being inappropriately sized in the first place).
To be clear, a typical Medium creature can wield oversized weapons as follows: a Large light weapon as a one-handed weapon, a Large one-handed weapon as a two-handed weapon, and a Huge light weapon as a two-handed weapon. The warrior suffers a –2 penalty on attack rolls with the Large weapons and a –4 penalty on attack rolls with the Huge weapon.
However, a typical titan fighter adds to this: a Large two-handed weapon as a two-handed weapon. The warrior suffers a –4 penalty on attack rolls with this Large weapon.
Further, the titan fighter special ability incredible heft reduces the penalties for (but doesn't grant bonuses to) a warrior employing a weapon one size category bigger than the warrior. In other words, it's only at level 19 when a Medium titan fighter finally can use a Large two-handed weapon without penalty.
Best Answer
Although the character's trying really hard, he still can't wield a too-big two-handed weapon in one hand
What stops the Medium titan mauler barbarian 3/titan fighter fighter 1 from wielding in one hand a Large two-handed weapon—despite his many special abilities allowing him to wield weapons of unusual sizes—is the restriction on the extraordinary ability jotungrip of the archetype titan mauler, which says
Emphasis mine. The titan mauler archetype's extraordinary ability massive weapons allows the warrior to use 2-handed weapons that are inappropriately sized more easily, and so does the titan fighter archetype's extraordinary ability giant weapon wielder, but neither special ability changes how the special ability jotungrip functions, and that special ability functions only on appropriately sized weapons, and that's the one that would need to change for the character to use in one hand a two-handed too-big weapon.