One is the Colour Phi Phenomenon, where two different coloured dots (with some space between them) are flashed in rapid succession. It appears as if there is just a single moving dot which abruptly changes colour as it passes the midpoint. That is, it seems to us as if we see the second colour before the second dot is even displayed! (If you follow the link you can try it yourself. It does not seem to work on me - or perhaps it is my computer's fault - all I see is the two coloured dots which I would describe as flickering back and forth. But I think that is best understood as a theoretical construct on my part - I only infer movement, I don't observe it. I don't actually see a dot in any intermediate position, so I can't see any early colour change either. But this is a well-confirmed phenomenon, I am told.)
Another remarkable result is known as the "cutaneous rabbit". Here the subject receives a series of taps on the wrist followed by taps at a position higher up the arm - with up to 200 msec between each tap. Remarkably, it seems to the subject as if the taps were spatially spread out across a sequence of equidistant points, "as if a little animal were hopping along the arm" (p.143). Of course, if they received only the wrist taps then there would be no such effect. But then how is it that we are conscious of the moving 'hops' before [it seems that] the upper-arm taps have even occurred? It must be that the later discriminations somehow influence our subjective experience of the seemingly-earlier time.
The most striking example of all (though also quite controversial, as it has never been successfully replicated) is an experiment by Libet. Recall that: (1) it takes time for sensory messages to travel from the nerves in our left hand to the neurons in our brain's right hemisphere; and (2) stimulating your left cortex appropriately can produce the feeling of a tingle in your right hand. Now, Libet stimulated a patient's left cortex before their left hand, so we would expect that they would feel a tingle in their right hand first, followed by their left. But apparently some gave the opposite answer, demonstrating a significant incongruity between the objective sequence of brain events and the subjectively experienced order.
Although these results at first seem very bizarre, the mystery disappears once we distinguish between the 'vehicle' and 'content' of a representation. Dennett points out that we can represent time using a medium other than time itself. You can say "B occurs after A", and it represents the ordering (A,B) even though the sentence mentions them in the order (B,A). Similarly, the brain can represent time using something other than time itself. You can have a represented order of events (A,B) in your consciousness, even if the brain regions doing the representing processed the events in the opposite order.
What matters for the brain is not necessarily when individual representing events happen in various parts of the brain (as long as they happen in time to control the things that need controlling!) but their temporal content. That is, what matters is that the brain can proceed to control events "under the assumption that A happened before B" whether or not the information that A has happened enters the relevant system of the brain and gets recognised as such before or after the information that B has happened. (p.149)
This explains how an incongruence between objective and subjective time could occur, but we are still left wondering why. Dennett suggests the answer is found by reflecting on the brain's "fundamental task":
The brain's task is to guide the body it controls through a world of shifting conditions and sudden surprises, so it must gather information from that world and use it swiftly to "produce future" - to extract anticipations in order to stay one step ahead of disaster. (p.144)
Given the time pressure the brain is under, it could well be economical to represent temporal information using something less valuable than time itself.
But what? Some form of "date stamps" would be one (theoretical) possibility, but Dennett points out an even cheaper (and more biologically plausible) method, which he calls "content-sensitive settling". The basic idea is that various representations are 'jiggled' around together until their contents cohere - Dennett uses the analogy of a film studio 'synchronising' the sound track with the film.
Lastly: an important consequence of this view is that internal representations have no exact 'moment' of occurrence - they are both spatially and temporally 'smeared' through the brain. An object's colour might be represented in some places and times, its location and motion in others. There is no instant when it 'all comes together' in consciousness.