I'm just now forming my companies identity, but I'm open from anything that is very modern and minimalist, to something that is aggressive with a lot of personality. Modern may make it more accessible to some, but I'm not worried about having a strong personality with my company. I grew up skating, so a lot of those logo designs influenced my taste. At the same time, I'd like for designers to give me something that is also their own style. I'd love to get a design that looked like it came from an artist, and not just what someone told them they wanted. Obviously an aggressive, angry Yeti with a video camera (or even a video camera for one of his eyes, or no camera at all) is one direction, but I'm open to just a silhouette that blends into the rest of the font/logo, or even avoiding the Yeti all together. I want designers to make this their own if they're feeling inspired.
If the design is illustrative, I would rather it be either aggressive or of a style that wasn't the typical looking shiny, cute creature/animal designs that are currently popular, and that look like they came out of an Adobe Illustrator tutorial. I know simple colors work for logos, but my preference (and personality) is more Ren & Stimpy than Curious George when it comes to style.
Below is just some art and colors I like. I know these are complex illustrations but hopefully they give you a feel illustrations that appeal to me. Obviously I want to appeal to possible clients and not JUST what I like, and anything vulgar in these aren't what I'm going for either.