The Large Hadron Collider accelerates particles to 99,9999991 % of the speed of light. And I understand that you need infinite amount of energy to accelerate it to the speed of light. What would happen if the Collider was vertical and you could use the force of gravity - would that change something? I understand that light has no mass and therefore no acceleration is possible - but the particle accelerated has a mass and therefore gravity should have an accelerating effect, or am I wrong?
Your reasoning is correct in theory, but flawed in magnitude. Placing the Large Hadron Collider vertically would have little to no effect on the final velocity of the particle. Since the particles fired through the collider are nearly massless, the potential energy of the particles is very, very small. The corresponding increase in final velocity will be insignificant compared to the final velocity in either case.
In addition, as Kyle Kanos points out, building the collider vertically would present some severe engineering challenges, which easily would wipe out any minimal gains in terms of increasing maximum particle velocity.
Since the collider is a big circle, even if you mounted it vertically it would make no difference, because half of the time, the particle would have to oppose the gravity. But the other answers are also correct in that the magnitude of any effect due to gravity is too small to make a difference anyway.