The app pool advanced settings allows you to turn on extra logging for the app pool. You can turn on all of the app pool recycle logging. It shouldn't be chatty in a healthy situation and you don't mind if it's chatty while you're troubleshooting.
Then in Event Viewer you should get an event logged every time your app pool recycles.
In IIS, an Application runs within an Application Pool. An Application Pool runs in one or more W3WP processes.
It sounds like what's happening here is that your app eventually encounters a condition that the W3WP can't cope with when run with minimal privileges.
LocalSystem is a high-privileged account - when stuff starts (or in your case, keeps) working as LocalSystem, typically that indicates some form of privilege level difference (occasionally, user profile differences).
All App Pool Identities are magically given membership to IIS_IUSRS by default, which grants all the permissions IIS requires to run an App Pool - this may not include all the permissions required by your particular app/framework/one of the libraries.
Try enabling the Load User Profile setting in the app pool when running low-privileged, and if that doesn't improve things, you'll need to debug it - either watch it running with Process Monitor for possible file or registry access difficulties (http://live.sysinternals.com/procmon.exe), or grab a memory dump of the process when it's failed, grab PSSCOR2, and perhaps start with !dae if nothing looks particularly odd at that point.
The short version is that the internal process / runtime state sounds like it's getting lunched; it shouldn't; often it's going to be some sort of native code issue that does that, but in your case, it could be permissions.
Edit: Oh, not LocalSystem, {local system user like administrator}. Gotcha. Well, give the Profile thing a try - see also this bit.
Also, Recycling: when a new W3WP is started to replace the old one, losing all the state from the old one in the process, and the old one is killed off after 90 seconds if it doesn't complete any useful work it's still capable of, and shut itself down. (read "by default")
Best Answer
On IIS 8.5 (Windows 8.1), I've just run the following test.
My start time is 11:04. I set the app pool to recycle at 11:06 (specific time), and also reset every 5 minutes (regular time interval).
Using PowerShell -
First, when checking the start date of the process, I get 11:04:27 (The time when I changed the settings of the app pool, forcing a recycle).
Then, I get 11:06:00 (when the specific time passes).
I let time go on. The next change is at 11:11:00. This is 5 minutes from when the last specific time recycle kicked in.
So, all in all, the regular time interval counter is reset when the specific time passes.
TL;DR
In the given question, the app pool will reset at 3.00am every day, because the 1740 minutes always gets cut short by the specific time recycles.