A temporary cache is kept at '%APPDATA%\The Classic Browser' while it runs but is allegedly wiped out upon termination. I'm not a Chromium user so I didn't test any of it but I haven't seen it mentioned around TPFC so here goes...
author of http://theclassictools.com/ wrote:The Classic[ Browser]'s dynamic cache-less nature and a host of other settings, ensure your 100% privacy, which you can easily verify with a 3rd party online privacy testing tool like 'Panopticlick' or 'Aloodo'. You'll find that the Classic is the only browser on the market that attains perfect privacy protection, straight out of the box! [...]The Classic Browser uses the Chromium engine strictly as a renderer-only and that's where the similarities with Chrome end. Our entire interface and all number-crunching routines are custom written from the ground up [...] Chromium is only called to pass a URL to it to display a website, in fact we have fine-tuned the highly configurable Chromium engine not only to support the latest web technologies but to also not waste precious time writing to your hard disk and thus the absence of Internet junk files.

A SoftPedia page and review can be found at http://www.softpedia.com/get/Internet/B ... wser.shtml.
The Classic Browser latest 32 or 64-bit portable release (currently v3.4) can be downloaded from http://theclassictools.com/.