Definition of general anesthesia
A state of unconsciousness does not guarantee analgesia nor amnesia. Besides, a muscle relaxant is often paramount to prevent movements. The perfect general anesthetic would do all of this and have hypnotic, analgesic, amnesic and muscle relaxant properties. Such compounds do not exist, because each effect works on a different part of the brain or elsewhere.  

In practice, therefore, multiple compounds are administered to obtain the desired effect. A cocktail anesthetic not only ensures all the desired physiological changes, it also reduces the chance of toxic side effects relative to giving one constituent in a large enough dose to encompass all the effects. And there lies an important point, namely anesthetic depth. 

If anesthesia is deep enough, all effects may, perhaps, be reached to some degree with a single compound, but the dose likely becomes so high that the side effects become too large. An example in this realm is alcohol, which when consumed in copious amounts, may come close in reaching all desired effects at least to some degree. A related compound is ether. But even with ether, which has been used in early days of surgery, surgical stress caused patients to die intra-operatively due to cardiac hyperactivity. Surgical stress needs to be minimized and that's why an analgesic is needed in the cocktail. 

Commonly applied anelgesics in the OR are opiates, such as fentanyl. Anelgesics are critical for invasice surgery, because the body reacts to trauma even when the brain cannot process the information to the degree necessary to turn it into a conscious experience. 

For instance, a widely used cocktail is propofol-based anesthesia: A propofol injection might be used to start anesthesia, fentanyl (injectable) is used to blunt the stress response, midazolam (injectable) given to ensure amnesia, and sevoflurane (inhalant) during the procedure to maintain the effects.