The Chicago Manual of Style, sixteenth edition (2010), offers the following advice, under the general heading "Multiple Punctuation Marks":
6.118 Periods with question marks or exclamation points. A period (aside from on abbreviating period; see 6.117) never accompanies a question mark or an exclamation point. The latter two marks, being stronger, take precedence over the period. This principle continues to apply when the question mark or exclamation point is part of the title of a work, as in the final example (cf. 6.119).
Their first question was a hard one: "Who is willing to trade oil for water?"
What did she mean when she said, "The foot now wears a different shoe"?
She owned two copies of Will You Please Be Quiet, Please?
The fifteenth edition of Chicago (2003) offers yet another example of this rule in action, at 11.35 (Question marks in block quotations):
The narrator then breaks in: "Imagine Bart's surprise, dear reader, when Emma turned to him and said, contemptuously, 'What "promise"?'"
The reason underlying this preference is *Chicago'*s notion that periods are weaker than question marks and exclamation points, and therefore must give way to them—and indeed vanish altogether in their immediate presence.
A very similar rule appears in The Oxford Guide to Style (2002) in the general section on Quotation marks, under the specific guideline 5.13.2 (Relative placing with other punctuation):
When the requirements of the main sentence and the quotation differ, use the stronger mark. In the examples below the question mark supersedes the weaker full point [period, in U.S. English]:
She was heard to mutter, 'Did you do it?'
Can you verify that John said, 'There is only one key to the room'?
The instances from Oxford that Mick points out in his answer, where an exclamation point and a question mark appear on either side of a close quotation mark, reflect situations where the competing punctuation marks are of equal strength or importance, as the text explains:
When the terminal punctuation of the quoted material and that of the main sentence serve different functions of equal strength or importance, use both:
She had the nerve to ask 'Why are you here?'!
Did he really shout 'Stop thief!'?
Oxford thus takes the view that question marks and exclamation points are of equal strength, whereas question marks (and by logical extension, exclamation points) are stronger than periods. In this respect Oxford and Chicago are in agreement.
From the foregoing guidelines and examples, it seems clear to me that Chicago would endorse this punctuation of your example:
"I have thousands of them," the commissioner answered, "but I'll start with 'What do you need?'"
and that Oxford would endorse this punctuation:
'I have thousands of them,' the commissioner answered, 'but I'll start with "What do you need?"'
Neither style guide would preserve the end-of-sentence period/full point.