Small wedge under the saddle that carries the puncture repair stuff. What you carry is up to you but the tube I wrap in a cloth before putting in a plastic bag. Saves some abrasion from the plastic after it has been sitting there for a years riding. Pump and get a good one. Others will have a favourite but Topeak Morph Series has a good following here. I have two. The Morph G that has a gauge on it but it is not the easiest to mount on a bike. It is long and takes a fair amount of space. The other is a "MiniMorph" and may seem a bit on the small side. The mount for it does take the space of a water bottle but i can manage that. But that "MINI" pump works. Air going into the tyre at each stroke seems small but it does not take long to get 125PSI into the tyre--and with ease.



Pic is of the pump on a bike so you can see how small it is. It also shows the wedge under the saddle.

Going Old School- They used to be called Frame pumps but is a Full size pump that mount by jamming along the top tube between the head tube and the seatpost. Good versions of these work well but do not fit every style of frame.