The most implausible part seems to be that L is completely anonymous. He is clearly operating with other people, therefore he leaves a trail of information. Even if he is perfect at covering that up, people working with him aren't.

Light could also use his powers to extort. Someone in the police/media/politics must know who L is. Therefore it would be only a matter of time until you hit a person that knows him. An alternate approach to this, which is contrary to Light's beliefs, is to keep executing important people until L is revealed, someone will reveal him eventually.