Spread would fit the bill. It's not a logging service but a number of projects use it for distributed logging (such as the Apache web server's mod_log_spread module). It has a client for C# available here.
Its value lies within a simple principle: applications that produce logs shouldn't be constrained or interrupted by the very act of logging. Logging is typically a secondary concern of an application, and having your log file destination disk fill up (or your log service crash) is a really bad reason for your application to fail.
To reach this level of separation, you must build your application in such a way that it can log completely asynchronously and independently of the logging "service". Spread achieves this using a well-understood and established technology: UDP/multicast.
Logging with Spread would mean that your applications would log in a "fire and forget" mode over UDP. If no Spread daemon has been set up to consume those multicast messages, the log messages will disappear forever. However, the decoupling between log producer and consumer means that your applications will never need to worry about logs filling up disks, nor will they have to worry about spending time finding and connecting to a central log server using TCP.
Spread's not for every application, though. Your environment might not support multicast (so EC2 is out, for starters) or your network admin might prevent multicast packets within your production environment.