A significant instance of hyperreality that I can document is the phenomenon of false awakenings in lucid dreamers.

Wikipedia defines hyperreality as:
"when a consciousness loses its ability to distinguish reality from fantasy, and begins to engage with the latter withoutunderstanding what it is doing, it has shifted into the world of the hyperreal"

The phenomenon of false awakenings occurs when a lucid dreamer (a person who can control what happens in their dreams) desires to leave a dream. They try to prime themselves to wake-up or will themselves to wake up. However, their subconscious tricks them and, while they are still lucidly dreaming (and thus in the same state they are when they are awake), they dream that they are actually waking up in their bed. They thus believe that they have woken up when they actually are still asleep. The dreamer then may then try to return to "sleep", however, often dream-elements (i.e. things that cannot possibly happen in real life) occur. When this happens, the dreamer may truly believe that they are awake and become very nervous, as they believe that they can no longer control what is happening. Often, even lucid dreamers do not use their potential to solve any problems that occur.

Because the lucid dreamer truly believes that the false awakening dream world is reality, this is an example of the lucid dreamer being in a state of hyperreality.