Why does economic growth in this example not increase the value of my currency? It seems very probable that the example is oversimplified, in which case I'm looking for a simple example that nonetheless captures how economic growth leads to a currency appreciating.
Because you are de facto on fixed exchange rate.
Under flexible exchange rate value of a currency depends on multiple factors. Let just take simplistic monetary model of exchange rate given by:
$$\ln S= \ln(m)- \ln(m_f) -(\ln(y)-\ln(y_f))+\lambda(i-i_f)$$
where $S$ is the exchange rate $m$ is the money supply, $y$ real output, $i$ interest rate and $f$ denotes foreign country (variables without $f$ are home country variables). This is not necessary even the most accurate model since we can add expectations and things like scapegoating in and so on. So in real life flexible exchange rate depends positively on real economic output but also on multitude of factors unrelated factors that can offset the effect of real economic output. This is what the other two answers are talking about.
However, you are de facto on fixed exchange because of your assumptions. Exchange rate can also be expressed under the law of one price as: $S= P/P_F$, where $P$ is price level. In real life, law of one price can be violated due to transaction costs, transportation costs, tariffs and so on. But in your example you ignore all such things so law of one price applies.
In your example, there is only 1 good so price level is fully determined by that 1 good. Next you decided to arbitrary fix both price in US and in your made up country. Hence you fixed the exchange rate purely by your assumptions.
In real life when country becomes more productive price level adjusts as well. Price level is inversely related to real output, e.g. standard New Keynesain money market equilibrium which determines price level is given by $P= \frac{M}{L(Y,i)}$ where $M$ is money supply $L$ demand for funds/money, $Y$ is real output, and $i$ nominal interest rate. So as you produce more $P$ should decline, and when $P$ declines exchange rate appreciates. However, you assumed this adjustment mechanism away in your example, since you just assume the price of the burger is fixed no matter how much you produce which is not necessarily accurate theoretically or even realistically.
Of course, if you just wilfully ignore adjustment mechanisms in other parts of an economy you can get result where nothing happens. This should not puzzle you, in your example output cannot affect exchange rate, because you simply assumed (even if unintentionally) that it can't have any effect on price level.
