In Hamlet, how many examples of revenge are there?
Great question! I am pleased that you have obviously understood that the theme of revenge clearly stretches beyond its central instance, which is quite clearly that of Hamlet's need to satisfy the Ghost of his dead father and gain revenge against his uncle, Claudius, for killing his father. It is important to realise that there are two other characters that are actually juxtaposed with Hamlet at various points in the play for dramatic effect and to highlight Hamlet's shortcomings or virtues who both are pursuing their own campaign of revenge.
First of all, Fortinbrass is trying to gain back territory that was taken from Norway and his father by Hamlet's father. He therefore is plotting a military campaign to seize back these lands and gain his revenge, as Horatio in Act I scene 1 tells us. The desire of Fortinbrass is expressed in the following terms:
...which is no other
(As it doth well appear unto our state)
But to recover of us by strong hand
And terms compulsative, those foresaid lands
So by his father lost...
Fortinbrass thus is out for revenge, and wants the lands that belonged to Norway back. At various points, Fortinbrass is compared to Hamlet to highlight Hamlet's procrastination and his failure to act on his need for revenge.
In the same way, Laertes is clearly paralleled with Hamlet. Both of them have lost their fathers to murderer and both want to kill the murderer and therefore gain revenge. The desire of Laertes to kill Hamlet, or "To cut his throat i' th' Church" clearly expresses his passion to gain his revenge and slaughter Hamlet the way that Hamlet slaughtered his father.
Thus I would argue that there are actually three examples of revenge in this story that we can look at and identify. It is important to analyse these fully and in particular look at the way in which these two examples of revenge help to highlight the central example of Hamlet's desire to gain revenge.
There are actually five. The first three are obvious. There is Hamlet's revenge for his father, there is Prince Fortinbras' revenge for his father, and then Laertes' revenge for his father, Polonius. But there are two more.
The exposition the Player King recites for Hamlet at the end of 2.2 has an irony because Pyrrhus' slaughter of King Priam is an act of revenge for the death of his father, Achilles. The images here are stark. The object of Pyrrhus' revenge should be Paris, son of King Priam, who killed Achilles, but Paris is already dead. Revenge though is unreasoned blood lust. Pyrrhus doesn't just kill the "Old grandsire" Priam, he made "malicious sport" in "mincing" Priam's limbs. This snippet of a play has preserved historically and as somewhat mis-remembered by Hamlet, shows Pyrrhus to be "hellish" like the "Hyrcanian beast" with "black purpose". In other words revenge is not an admirable endeavor. But through this all Hamlet makes no note of it.
The next revenge is a little more obscure. It involves Julius Ceasar, who is referenced three times in the play. And by coincidence he was the focus of of one of Shakespeare's recently written plays. A play that has these ominous words spoken by Mark Antony:
And Ceasar's spirit ranging for revenge
With Ate by his side come hot from hell
Shall in these confines with a monarch's voice
Cry 'Havoc' and let slip the dogs of war
Ate, pronounced Ah-tay, was the goddess of rashness and folly.
Legend has it that in Shakespeare's company of actors, the actor playing Julius Ceasar was the same actor who would play Polonius. Brutus and Hamlet would also share the same actor. So the undercurrent of the humor in 3.2 when Polonius tells us he enacted Julius Ceasar and was killed in the capitol by Brutus and then Hamlet offers that it was a brute part of him (Brutus) to kill so capital a calf there just drips with irony when the brute part of Hamlet in his rash lust for blood kills the "unseen good old man" and father of Laertes.