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It was hard for me to gauge what percentage of total debt that is, so a quick search indicates that by end of year 2018 the US government debt is going to be 21.4 trillion.

Roughly 20% of US government debt is held by itself.

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According to the national debt's Wikipedia page 5.7 trillion is owned by the government. That ends up being about a forth of it.

I believe it's more than that. The Social Security trust fund holds another ~$3 trillion in federal debt.

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