CatLog shows a scrolling (tailed) view of the Android "Logcat" system log, hence the goofy name. It also allows you to record logs in real time, send logs via email, and filter using a variety of criteria. It has a widget, can record logs in the background and has useful colour coding.
It won't work on Android 4.1+ without root access, since apps are not allowed to access each others' logs.