OK, I think I see what the pattern is.
- Browsing to http://www.google.com, and entering my search terms in the search field, always gives a result page with a search field
- Entering my search term in Chrome's address bar and pressing Enter gives a result page with a search field
- Entering my search term in Chrome's address bar, and selectinga previous search or a Google suggestion, gives a search result page without a search field
Note that for scenario 2 and 3, it is not easy to distinguish between entering a search term and pressing enter, and entering a search term and selecting from the suggestion list. Pay close attention to which keys you are pressing, and to what is selected when you press Enter. In order to invoke scenario 2, I must press Enter immediately, before the suggestion list pops up.
It also seems to matter if I'm already displaying a Google search result page. If so, the next results page seems to always include the search field.