There have been some good suggestions about potatoes, but no-one has suggested
that you grow in containers, or tires, or anything similar. Containers with
suitable drainage holes can be 1/4 filled (approx) with compost and the seed
potatoes planted in this. As the potatoes grow, earthing up can be carried out
gradually until the plants have reached well over the top of the container.
Tires can be stacked one on top of another gradually over an extended period
until the desired height is achieved. At harvest time the containers can be
emptied and the potatoes easily harvested (at least more easily and efficiently
than digging up out of the garden).

I've assumed a certain amount of common sense will be used when choosing a
container, and also with regard to using an appropriate soil/compost, having
suitable drainage, choosing an appropriate potato variety to grow, etc.

There are a few distinct advantages with this system. Several containers can
be used and crops kept seperate to avoid cross contamination of pests and
diseases. The containers can be protected to keep out some pests such as slugs
and snails by attaching copper strips around the containers. The containers
can be started off in a greenhouse if desired, or be moved inside in case of
severe frost. The compost can be used for general garden use after the crop
has been harvested so the next years crop will not suffer from any
pests/diseases being carried over. (Next year you use new/fresh compost/soil.)
Containers often heat up more quickly than the soil in the surrounding area.
Stacked tires warm up particularly well when exposed to sunlight.

I hope this is of some use.

Dave.