Which electrodes do not corrode at all?
This highly depends on what you are electrolyzing. When using solutions that are not very acidic, oxide-coated metals sometimes work, like lead-oxide electrodes, that can be harvested from old lead accumulators (attention, lead is toxic, wear lab coat and gloves when working with them. Lead accumulators contain ~20% sulfuric acid, so wear gloves as well and unassemble them in an acid-proof working space)

If you search the web for recipes of perchlorate preparation by electrolysis at home, you will find extensive discussions on the topic of anode materials and making oxidation-resistant electrodes. 

Platinum is really hard to corrode, but even it corrodes in some cases.