The level cap for the Destiny alpha, which launched during E3 week exclusively for PlayStation, was level 8 (although players got a glimpse of what they could expect from higher levels via teases). That same level cap also holds true for the Destiny beta, which launched yesterday on PSN (a beta for Xbox will be coming soon, at a later date)- you can’t progress beyond level 8. If you try, your progress bar at the bottom of your HUD ceases to have a level number on its right side.
People who invested heavily into the alpha might be disappointed at essentially having to replay the exact same sections that they already went through just a month ago (at least in terms of the level curve/scale), but on the whole, it also appears that this is a good way to give players a feel for what the game will be like without actually spoiling too much of the game.
Destiny launches later this year for Xbox One, Xbox 360, PS3, and PS4.