Site too large to officially use Google Analytics?

Stackoverflow.com does blow well past that limit doesn't it! Fortunately, per paragraph 2 of the terms of service...

FEES AND SERVICES . Subject to Section 15 herein, the Service is provided without charge to You for up to 5 million pageviews per month per account, and if You have an active Adwords campaign in good standing, the Service is provided without charge to You without a pageview limitation.

... as long as you have an "active Adwords campaign in good standing", then the service is provided "without charge to You without a pageview limitation".

EDIT:

As of October 2011, the terms of service have been updated such that the page view limit was raised from 5 million to 10 million page views per month HOWEVER the AdWords exemption has been removed:

FEES AND SERVICES . Subject to Section 15 herein, the Service is provided without charge to You for up to 10 million pageviews per month per account.

Consistent with this answer here it is not exactly clear what the intent is for sites that exceed the limit and if/how the limit is really enforced.

For reference, the current full terms of service are documented here:

http://www.google.com/analytics/tos.html

I think that for being free, getting data updated once a day is just fine. I don't even know why with such large numbers you need more then that. Did you really lose anything?

If you really want alternatives, I suggest you try http://piwik.org/. I have no idea if it can take the load, but since it is up to you to install and configure, I guess all is possible (only questions is, how much work?)

Other hosted option is http://chartbeat.com/ which since you actually pay for the service, I don't think they will have a problem accepting your money.

Update: based on comments, http://www.getclicky.com/ is not an option.

Or just check results once a day. Is your site/service dependent on reports multiple times a day?