Yes. There are limits to what you can place in an AMP site. Basically, its all static assets, so the entire AMP version is supposed to be cached by Google.
Does this means there is no way to opt out of connecting to a Google-controlled server in order for any AMP-based page to display properly?
> What gets cached?
> If an AMP page is valid and is requested (so the Google AMP Cache is aware of it), it will get cached. Any resources in AMP pages, including AMP images, also get cached.
> Can I stop content from being cached?
> No. By using the AMP format, content producers are making the content in AMP files available to be cached by third parties. For example, Google products use the Google AMP Cache to serve AMP content as fast as possible.
> Can the cache be crawled?
> No. The Google AMP Cache is roboted to crawlers. We recommend that search engines process cache links according to the guidelines for crawling Google AMP Cache URLs. For more information, see robots.txt."