When painting your house, make sure you have a damp rag with you so you can immediately clean up any drips or mistakes. If the paint you are using is in a traditional metal can, you'll need a flat-head screwdriver or other tool to pry off the lid, and a hammer or mallet to close the lid tightly. If you have these tools ready when you begin your painting job, you won't have to interrupt your work to go fetch them later.
I would say a roller, a paintbrush to get the corners that rollers can't, and a pan to pour the paint into so you can roll your roller onto it. You can live without newspapers, tape or screwdrivers to open the lids, because those items are easily replaceable with other items. But there's no way you can paint without rollers, brushes or pans.
1) A clean rag.
2) A 5-way or stiff putty knife.
3) Duster brush.
More is needed depending on you are doing but these three are needed in or out and doing almost everything.