A composite event occurs when all (or many) of a doll's imprinted personalities and skills reassert themselves in his or her brain. Such an event was considered theoretically impossible, since each imprint is wiped completely from the doll's brain after each engagement. However, such an event did occur with Alpha, due to an accident with the imprinting chair.
After a composite event, it is hard to tell who is the dominant personality. Also, as evident with Alpha, a doll who suffers a composite event is not only aggressive but very intelligent as well. Alpha has proven this on multiple occasions, with the most notable incident being when he hacked the Dollhouse secure network and performed a remote wipe. This is impressive, not only because the Dollhouse has a great amount of security, but because remote wiping was supposed to be impossible. This would mean that a composite doll can access the combined knowledge of their imprints, or the imprints have become direct parts of their personality.
The first known composite event occurred in 2008, when an accident in Topher Brink's Imprint Room caused at least 40 personalities to imprint at once. This appeared to cause Alpha to go insane (or at least allowed his psychotic original personality, Kraft, to become dominant), while at the same time becoming exceptionally skilled in combat, technology and survival. This combination made Alpha extremely dangerous, to the point where he was able to kill a Handler in a period of eight seconds, when the injuries he inflicted should have taken hours. (S1: "The Target", "Briar Rose", "Omega"; S2: "A Love Supreme", "Epitaph Two: Return")
Echo's composite event occurred in 2009, when Alpha kidnapped her, and imprinted at least 36 different personalities, except for Caroline Farrell, into her. This started a chain reaction that eventually allowed Echo to resist imprinting and wiping if she so desired, and allowed her to switch between retained personalities at will. Unlike Alpha, Echo did not lose her sanity, but she subsequently suffered severe headaches that gradually became worse each time a new personality or skill set was imprinted. (S1: "Omega"; S2: "Vows", "Meet Jane Doe", "A Love Supreme", "Stop-Loss")
Who has experienced a composite event
After attacking Whiskey, Alpha had all his previous personalities imprinted into him at once due to an accident in the Imprinting Room, causing a composite event. As a result, Alpha has become a highly aggressive and intelligent being who is nearly impossible to trace. After the event, his first act was to destroy his original personality, presumably for 'abandoning' him. His skills are many and he seems hell-bent on taking the Dollhouse down.
In "Omega", Alpha abducts Echo and engineers a composite event in her, similar to his own. However, Echo does not show the psychotic tendencies exhibited by Alpha (which were probably caused by the remnants of his original personality), and refuses to kill Caroline (who was imprinted into an innocent bystander). She eventually returns to the Dollhouse and gets wiped, apparently returning to her former status. At the end of the episode she whispers the word "Caroline" to herself, which could be either another sign of her growing self-awareness (which had already started happening long before the composite event) or an indication the wipe was not completely successful after the composite event.
Echo has shown signs of a growing self-awareness and personality, though whether these are symptoms of a natural composite event or her own evolution remain to be seen. In "The Target", exposure to drugged water caused Echo to see past imprints, meaning she could very well be close to a spontaneous composite event, and the right stimulants are the only thing now needed to trigger it. Interestingly, during the events of "Briar Rose", Echo seemed to show signs of compositing when she remembered bits of her imprints and acted accordingly, fighting Ballard in self-defense when she remembered their battle before, and recognizing Boyd as a friend when the imprint bond should have been wiped.
In later episodes, Echo is showing more and more retention of her previous imprints, and in "Epitaph One", she remarks to Paul Ballard (who would shortly become her handler, early in the second season) that all of her imprints are active in her mind, and that this causes her to have extremely painful headaches. Echo's multiple imprints in "Epitaph One" and her compositing in "Omega" show that her concurrent imprints are more cooperative with one another than Alpha's, or that Echo herself is better able to control them.
According to Boyd, Caroline Farrell, aka Echo, was born with a specific gene which allows her to resist the wipes and evolve in her Tabula Rasa state.