I did a test today and if you start a PC in Safe Mode with Networking and you go to the Services application, if you try to right click and start the "LogMeIn" process, it will say it can't be started in safe mode. Bummer.
Thankfully, there is a solution for that - you add the service to the trusted list of processes for Safe Mode, and presto, you can start it.
You just need to add a key in the registry using this rather obscure command (start a new command prompt), and you are almost done:
reg add "HKEY_LOCAL_MACHINE\System\CurrentControlSet\Control\SafeBoot\Network\LogMeIn" /VE /T REG_SZ /D "Service"
(one line). The output should be like in the next picture
Now, the process can be started in safe mode (with networking) by right clicking it in Services and selecting "Start". Don't worry if the process takes a long time to start or if it generates an error saying that it takes too long to start. Even in this crippled state you can log on remotely and do your thing.
This trick is persistent across reboots. The next time you get in Safe Mode, LogMeIn will be started on boot (bootup may be slower, though). Beware, there is a risk - if for some reason LogMeIn is the cause for Windows not starting in the Normal boot, it will also hang in Safe Mode with Networking!