This is what I (usually) take (I say "usually" because sometimes I swap equipment between bikes then forget to put it back on the main bike)...
for short rides around town
inner tube
mini air pump
co2 with working nozzle
two tyre levers
portable rain jacket
for longer rides (or even shorter rides but in the sticks), in addition to above:
another mini air pump
small bag of sweets or nutrition
chain links
puncture patch kit
derailleur hanger
2 x £10 notes
Front and rear lights and a bike computer goes on most rides too along with two water bottles.
Why take two pumps? Chance of puncture, co2 failing AND one pump failing is probably up there with Boris telling the truth!
Assume you also take a multi-tool?