Keychain style flashlight or headlamp, LED type, not incandescent: The keychain style is tiny and lightweight, the headlamp style is a bit bulkier but easier to use. You can get these for a couple of bucks or less at the local big box hardware store. The LED ones are lighter, more durable and last considerably longer. Very useful if you need to make a repair after dark! I keep a keychain style one as a backup as I normally have a headlamp in my backpack.
If you are planning on going on a multi-day ride or just riding a long distance in a day, you should have something for first aid, just in case. Unfortunately, almost everyone who cycles frequently will go down at least once in their life. This kit can be really small and compact and can include bandages, some gauze, and a small bottle of a disinfectant.
Broadly speaking there are two major wheel sizes – 700c for road bikes and 26 inch off-road wheels. However there are a whole variety of older, less common wheel sizes and of course fold up bikes have much smaller wheels altogether. Repairing a puncture is not an emergency! It’s a basic procedure that every cyclist should be able to to perform, but doing it correctly involves more steps than many people realise. Wheel building is often seen as the most artful skill in bicycle maintenance, and separates the professionals from the amateurs but that doesn’t mean you should consider it out of reach. Truing a slightly misaligned wheel is a cycle maintenance job that almost any cyclist can perform.