Probably one of the best ways to be successful with a new logo is to really think it through. Making anything, just to get rid of it won't make you a good designer. You might think you gotta go with the standard logo: graphical element + logotype, but you don't have to, actually.
A logo is a graphical element (ideogram, symbol, emblem, icon, sign) that, together with its logotype (a uniquely set and arranged typeface) form a trademark or commercial brand.
That said, one element logos are the ones that got one or another. These are some great exemples of successful logos that achieved their objective by solving their problems with creativity. It's not necessary to have an icon supporting your logotype... or a logotype supporting your icon. If you find the right balance in a single element, then most likely you're good to go! So, here are some amazing one element logos that I collected from amazing Logopond. I hope you enjoy these! Cheers!