Google doesn't actually store the text that it searches. It stores search terms, links to the page, and where in the page the term exists. That data structure is indexed in the traditional database sense. I'd bet using wildcards would make the index of the index pretty slow and as Developer Art says, not very useful.
Google does search partial words. Gmail does not though. Since you ask what's the problem here, my answer is lack of effort. This problem has a solution that enables to search in constant time and linear space but not very cache friendly: Suffix Trees. Suffix Arrays is another option that is more cache-friendly and still time efficient.
It is possible via the Google Docs - follow this article:
http://www.labnol.org/internet/advanced-gmail-search/21623/
Google Code Search can search based on regular expressions, so they do know how to do it. Of course, the amount of data Code Search has to index is tiny compared to the web search. Using regex or wildcard search in the web search would increase index size and decrease performance to impractical levels.
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