In the Haftorah for parshas Zochor we read that Hashem commanded King Shaul not only to destroy all the Amalekites, but also all their animals. Why was there a need to destroy also the animals?
Rashi to Shmuel Aleph 15:3 explains that the Amalekites were sorcerers and were capable of disguising themselves as animals - and for this reason Shaul was commanded to kill even the animals.
In his commentary to Devarim 25:19 he brings another explanation: The eradication of the memory of Amaleik had to be absolute, and even if animals remained alive they would serve as a reminder of the nation. Every last animal had to be wiped out in order to ensure, "That the name of Amalek should never again be mentioned, even regarding an animal, to say: 'This animal was from Amalek.'"
If we look at 1 Sam 30:16, we see that the Amalekim, having taken the spoils of war, feast and drink and party like there's no tomorrow. In chapter 14, we see the soldiers of Israel also acting in an unbecoming way with the spoils of war. There, Shaul pronounced a curse on any soldier who eats anything during the battle. Of course, being soldiers, they get hungry, and eventually end up killing and eating the animals of the Philistines without properly koshering them (see verse 32). Now we come to chapter 15, and it's no surprise that Shaul is commanded to kill all the animals. Israel has just behaved like Amalek, and so this time around (especially when facing Amalek themselves), that cannot be allowed to happen: all the spoils of war must be destroyed.
(this idea comes from a local Tanakh teacher)