How can I bleed my hydraulic disc brakes without buying a kit? Is it possible to bleed hydraulic disc brakes without buying a kit?  If so, how would I go about doing it?

I have Shimano BR-M355 brakes, but the question is also applicable to other manufacturers.

What would I need to be able to bleed these brakes, and how does the end result compare?  In the UK a Shimano kit is approx. £30 [Sept 2016] including brake fluid.
It depends on your brakes.  

Some models need nothing more than a couple of short lengths of plastic hose, a cloths peg and syringe. Others need fittings to attach the hose(s) to the brakes, and you may or may not find something else to do the job in a hardware shop.  

Kits are mostly relatively cheap and probably cost no more than individual bits purchased  from a shop. Some kits are expensive (Avid comes to mind) - probably because you have no choice and have to buy it for the proprietary bits needed to do the job.