So I started monitoring things — and yes, it the server crashed every 3 hours as exponential amounts of memory were being reserved until the machine crashes.
I had a hard time isolating the issue as it seemed to stem from index.php (which is unusual), so I did the usual thing and cleaned out some large-ish values from the wp-options table and installed memcached.
Then I contacted the guy who had done this accounting software integration, who rambled something about a cron job.