What can I plant in a very narrow border which requires little maintenance?

I'll have to give it more thought, but Berberis "Helmond's Pillar" springs to mind immediately. Presumably you don't want plants that get too wide and block the drive, so columnar shape probably best. What about bamboos? Though with these you might have to block their root access under the tarmac by putting in a barrier 18 inches down - that way they'll only go sideways, hopefully. Maybe more later.

I would plant ornamental grasses. The come in so many varieties and colours. I have them in my front garden. Will take photo and submit soon. The best thing is they are low maintenence and can be divided.

maybe box hedging though it needs a trim

Did you want to hide the fence and have privacy? is it a solid fence or a wire fence?. I think this would determin what to plant to help fullfil your requirements. Also not wanting to have to clip or trim the other side if it is only a wire fence.

I have a border of similar width - I have planted it with small hardy geraniums and grasses plus some dwarf gladioli

what about delphiniums , there are a range of heights. red hot pokers penstemons, phystegia [obedient plant] are also suitable for mid height.

we have a similiar space though not as long and have planted a box edge, rosa wild edric which is really dense and flowers for a long time, a couple of lavenders which dont seem to mind facing east and underplanted with tulips and daffs. These are all perennial, so if you want same all year round I would go with the box hedge or shapes and plant bulbs