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I'd really like to know how much money has NASA administered since its inception.

Just a ballpark figure, hundreds of trillions?

Not even close:

    According to the Office of Management and Budget and the Air Force Almanac,
    when measured in real terms (adjusted for inflation), the figure is $790.0 billion, or
    an average of $15.818 billion dollars per year over its fifty year history.

I think you're a few orders of magnitude off. ~$20 billion per year for ~50 years is closer to $1 trillion. Just a guess though.

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