What makes a battle epic, as I believe, is its essence and purpose. What is the cause of the fight? What are the circumstances that await? How many people or lands will it affect, and will its impact be phenomenal? A Pokémon battle, for instance, is such a minor fight compared to something else, such as the battle at Minas Tirith (like RL mentioned), but it still has a purpose, which is actually to KO someone else's Pokémon and pawn them XD Then, there's revolutionary battles, such as forces rising against forces stronger than themselves. For example, in my "Alice," Alice Liddell is a girl fighting for the inhabitants of Wonderland, for the wicked Queen of Hearts has enslaved them and morphed the mad world into a realm of insane torture. In the end, Alice and the Red Queen will fight, and whatever the outcome is, it will affect Wonderland forever. If Alice wins, everyone will be free, and peace will be restored. If the Queen wins, Alice dies as her parents did, and Wonderland will be pressed under the Queen's thumb with more pressure than the last time.
So you see, it's mostly what will become of everything surrounding a battle, and the change that battles fight for, IMO.
EDIT: Seijiro's right as well. Big foes equal more fear and the dread of trying to overcome such a crushingly powerful force. The main character, or the hero, is fighting for a cause, and to place more twists than good defeating evil, will have suffered great losses to get that far, rather than their own life. Their survival until the final showdown is what makes their entire bodies pump with adrenaline, that they've crossed boundaries and succeeded in coming over things they didn't think they could. The fate that is sealed on the outcome of a battle will make their heart race, and the past that drove them there is an epic itself in a sense of accomplishment.