what are some energy requiring activities in a cell?

Answer 1

With the exception of passive transport, every cellular activity (from mitosis to DNA replication to respiration) requires energy.

http://en.wikipedia.org/wiki/Passive_transport

Answer 2

Energy is needed to:
1. grow by adding cytoplasm and organelles
2. move materials through the plasma membrane by active transport, forcing the materials to move from lower concentration to higher concentration
3. build molecules such as RNA, proteins, lipids, ...
4. package and export cell products
5. break down old organelles or macromolecules
6. build structures such as spindle fibers, and manipulate the spindle to move the chromosomes around during mitosis or meiosis