considering songs that were covered or remade by a lot of other artists ... as well as songs which were covered by artists in other genres.

With that criteria, I would put forth these songs:

'Ramblin On My Mind' & 'Stop Breakin Down' - Robert Johnson
'Yesterday' & 'Eleanor Rigby' - Beatles ('Eleanor Rigby' has been covered over 130 times -- surprising, I know)
'Cry Me A River' - Julie London
'(I Can't Get No) Satisfaction' - The Rolling Stones
'Imagine' - John Lennon (I hate this song, but it has to be in here)
'Over The Rainbow' - Judy Garland/Wizard of Oz
'Summertime' - Abbie Mitchell/Porgy & Bess
'Hallelujah' - Leanard Cohen
'Look of Love' - Dusty Springfield
'All Along the Watchtower' - Bob Dylan

There are plenty of others which should be on this list, but I can't think of them right now.