It needs to be big enough to contain the chair as you pull it out to sit down--you don't want it bunching up. It wouldn't have to go all the way to the wall under the desk.

On the other hand, it might be better to go ahead and get a large size to contain the desk, chair and the furniture you intend to put on the other side. Perhaps to cover the entire center of the room to a foot or two from the walls.

What size is the room?

The room is 13x16 - that would be a big rug if I got it for the whole room. I don't think I would want to go bigger than a 9x12 rug. Hmmmm

That might work--you would have 2 feet from the rug to the walls on all sides. The furniture could be arranged with only the front legs on the rug.

bumping up

The floors are pretty so not sure you want to cover them up that much but agree you need something bigger than you have now. Had you considered moving the desk so you are looking out into the room instead of at the wall? The bookcases would still frame/anchor the desk with that arrangement. Or you could have the light at your back and put the bookcases on either side of that one window and still be facing out into the room.

If any of those alternate solutions appeal to you, I'd try that first without a rug and see how the rest of the room shakes out for space. Unless you are trying to sequester the desk area from the rest of the room?

We could move the desk to face out - all the electrical cords printer router etc are connected in that corner so that is why the desk is there.

I just want the room to look as finished and decorated as possible since we can't purchase furniture for a year or so to complete the room. I thought curtain panels and a rug and some wall art/frames would look good.