There is absolutely no reason to take exactly 3 levels of Fighter.
Strictly Superior Alternatives
Assuming you go by non-fractional saves/BAB – which I refuse to do personally, but it is the base rule – the level gives you ~5.5 HP, +1 BAB, +1 Ref, +1 Will, and 2+Int skill points.
Taking Barbarian 1 at that point instead would get you ~6.5 HP, +1 BAB, +2 Fort, 4+Int skill points, and Rage, which effectively grants you +2 Will, +4 Strength, +4 Constitution. You also get +10 ft. movement speed, which is more minor. In effect, you are trading +1 Ref for: +1 HP, +1 Will, +2 skill points, +10 ft. movement, +4 Strength, +4 Constitution. Yes, you can only Rage once per day, but Extra Rage (Complete Warrior) is a good feat, and even without that Raging once per day is enough for me to favor the Barbarian here.
And you don’t even have to worry about multiclass penalties – another rule I despise – because Fighter 2 and Barbarian 1 are within 1 level of each other and the multiclass penalties don’t come into play.
With sane (read: fractional) saves and BAB, Fighter 3 gives +½ Fort, +⅓ Ref, +⅓ Will – Barbarian 1 gives +2½ Fort, +⅓ Ref, +⅓ Will to make it strictly superior to Fighter 3. It’s pretty common to houserule that you cannot get the +2 to a good save more than once per save, especially with fractional BAB, but in this case Barbarian 1 is literally equivalent to Fighter 3 plus 1 HP, Rage, and Fast Movement.
This is the purely-Core answer. If you add more supplements, there are far more answers available as options. If full-BAB isn’t strictly necessary, there are even more options: Barbarian 1 becomes better when you can get Pounce instead of Fast Movement (Lion Spirit Totem, Complete Champion). Cleric 1 is the best dip in the game (Domains/Devotions, spells, Turn Undead/Divine Feats). Crusader, Swordsage, or Warblade 1 are excellent (though best to take at ECL 5 if you can). Psychic Warrior can get you another Fighter Bonus Feat (or Psionic Feat), and also gets you some powers (like expansion). And so on.
Obscure Benefits of Fighter 3
I will point out that it is possible to get class features at Fighter 3 – the Zhentarim Soldier substitution levels from the Champions of Valor Web Enhancement grants Skill Focus (Intimidate) for free at Fighter 3. Doesn’t cost you anything, either. But +3 to Intimidate doesn’t really change matters in terms of which is better between Fighter 3 and Barbarian 1. Properly using Zherntarim Soldier means going to Fighter 9, for the ability to Intimidate as a Swift action.
I’m not aware of any great feats or prestige classes that require Skill Focus (Intimidate); if you found one, that might be a reason to take Fighter 3, but I’d certainly be loathe to do so. With all you can get from a dip in another class, I’d rather just take it as my normal feat. You already have two levels in Fighter; if you’re still that desperate for feats, something is very, very wrong. And for that matter, you might be able to get Skill Focus (Intimidate) from the Otyugh Hole (Complete Scoundrel); 3,000 gp for the opportunity to take something other than fighter for a level is a steal.
Additionally, the Dead Levels Character Class Companion allows fighters to get +1 to Strength checks to break inanimate objects, +1 to Dexterity checks to balance (but... not Balance checks? The wording is really unclear), or +1 to Constitution checks to continue running or to avoid damage in a forced march, at each odd level except 1st. So yeah, +1 to your choice of a single very-niche ability check. One of which doesn’t really make sense in the rules. Enjoy.
And that is literally it as far as things that affect or key on Fighter 3. No other ACF, substitution level, or whatever else affects Fighter 3. And even if one or both of these give you a good enough reason to take 3 levels of Fighter, as Matthew Najmon points out, at that point you probably want to take the 4th level, so even these are not good reasons, by themeselves, to take exactly three levels.