One can suggest many different things about the sequence of battle in the epic Beowulf. First, one could state that the battles represent the maturing of Beowulf. Second, one could state that the sequence of battles represents different types of internal battles within Beowulf. Lastly, one could suggest that the sequence of battles represent the emergence of Christianity over Paganism.
In regards to the maturing of Beowulf, one could suggest that Beouwlf's battle with Grendel illustrates his youth. Given his lack of real experience, his entering into battle without a weapon illustrates the confidence he possesses regarding his youth. The battle with Grendel's mother illustrates a more mature Beowulf. He takes a weapon into battle, thinking about the danger of the situation. In his final battle, Beowulf has ruled his kingdom for 50 years, and the dragon has lived for a very long time as well. Each has lived a good life, protecting what what theirs. Therefore, the battle with the dragon represents Beowulf's final battle and end of his life.
One could also say that the battles suggest different inner struggles with Beowulf. In his battle with Grendel, Beowulf must prove himself or let down Hrothgar and his name (which Unferth would love). In his battle with Grendel's mother, Beowulf must prove his resilience. Many of the men left Beowulf for dead. He needed to prove that he was the epitome of the hero. In the battle with the dragon, Beowulf needed to prove that he was still a hero (as an old man). He knew that a true hero lived and died for the battle.
Lastly, the battles could be seen as representing the epic battle of good and evil. Grendel's son illustrates the secondary demons of the world. Grendel's mother represents the primary demons. The dragon could be seen to represent the devil. Each battle gets harder and harder as it gets closer to the true villain--Satan and Paganism.