I just checked the US national debt clock and found out that every single American citizen, at birth, owes the government $208,668.
What did the average American citizens achieve for that amount of debt?
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What you saw was the debt per capita, not what each citizen owes to the federal government. There is no explicit provision in the U.S. Constitution for the federal government reclaiming its debt from its citizens. If the government tried to do this, it would simply not work. That figure you cited is roughly four times the median household income in this country, and I guarantee we would have close to 100% turnover in the U.S. Congress next election.
What did the average american citizens achieve for that amount of debt?
There are two ways to approach this.
First, look through the U.S. Constitution for a list of powers the government has. Specifically, Article I lists the types of laws that the U.S. Congress may pass. Some specific examples called out there are establishing a postal service, roads, army and navy, and regulating interstate commerce.
Second, look at the budget. There are many resources out there showing government spending, and it is a bit more complex than it looks at first. Here is one such breakdown of the budget. First, the budget covers interest on the debt. Next, it breaks down into mandatory and discretionary spending. Mandatory spending is mostly social security (government-run retirement for all citizens) and Medicare (government-run medical insurance for the elderly). Discretionary spending is broken down into quite a few categories, primarily military, government overhead, with others like education and veterans' benefits taking smaller pieces.
It is worth noting that here in the US, social security is paid via a separate payroll tax than the regular federal income tax. However, the fact that it is separated like that does not mean much considering it comes out of our paychecks and goes to the same federal government, who then spends it on checks written to people currently receiving the benefit.
That was the budget, not the debt, however. Every year, the government has a budget. That budget generally includes more spending than income, resulting in a deficit (there have been surpluses, even in the past 20 years, but these are rare in modern times). As the years pass, that deficit piles onto the debt.
In general, the US debt has bought its citizens whatever has been in its budgets. However, there have been other spending items added into the debt that politicians like to sneak in outside of the general budget. For example, the Iraq War in the 2000s was largely funded by supplemental budgets that increased the debt without impacting the budget per se.
The price tag has drawn criticism not only because prewar projections by the White House were closer to $50 billion, but because of the manner in which the bill was budgeted: through supplemental requests, often with little time for congressional oversight or full disclosure of how the money is allocated.
Source: The Cost of the Iraq War
The answer to your question is "the US public gets pretty much everything the government spends money on for its debt."