The recent injection of $2 Trillion into the US economy has prompted me to wonder: At what point does introducing money to a system cause hyper-inflation? (such as Venezuela or other hyper-inflation economies)
There is no set tipping point. In simple terms the relationship between money supply and inflation is given by the standard monetary model which shows that:
Where M is the money supply, V velocity of money P the price level (and change in the price level gives you the inflation) and Y is the real output.
Hence to get any inflation the changes of M must be such that they are not offset by changes in other variables.
If money supply changes by 10%, velocity of money stays constant, but real output also increases by 10% then there will be no change in prices and hence inflation will be zero. However, in the same example if the real output would not change and all other things remained as above the inflation would be 10%.
In normal times it is valid to say that inflation is a monetary phenomenon and depends mainly on quantity of money. The economy usually grows at steady rate of about 1-2% per year (I am talking about advanced economies at the technological frontier) and velocity of money does not change that much so in that case it is valid to say that relatively large increase in money supply causes inflation.
However, in situations like when interest rates are at 0 lower bound the velocity can actually easily offset any increases in money supply and relationship may break. Moreover, more complex models show that its expectations what matters. If money supply increases today but people expect this increase will be reversed in the future it might not affect inflation (you might want to look at this paper from Paul Krugman if you want to know technical details about that).
To go back to your main question:
At what point does introducing money to a system cause hyper-inflation?
Following Cagan hyperinflation can be defined as an increase in price level exceeds 50% per month.
To get to that rate money supply would have to growth at the rate of 50% + the rate at which economy grows per month + at any rate that would offset changes to velocity in of money. Special conditions like no zero lower bound interest rates and assumption that people credibly believe the money supply increase is permanent (or at least long lasting) must be imposed as well. In US there would also be special issues since \$ is the world’s reserve currency which helps to reduce inflation pressure as portion of the US money supply is absorbed by other countries.
So to sum up, there is no magic rate of increase in money supply. Even if money supply remains the same but output collapses you can get inflation. Alternatively even large increases in money supply might not change inflation levels at all as recent Great Recession experience shows. You have to do some complex econometric analysis to see at which rate in a given year money supply increases start to affect prices. Since that requires some historic data it’s not possible to say for sure what the level is at a given point of time.
You can make forecasts of course and central banks do that routinely. This is the FED inflation forecast for 2020 which is also based on money supply changes (aside from other things). Although it does not mention money supply explicitly because FED does most of the operations through the funds rate they factor it in their models (the money supply also expands and contracts based on the funds rate). They do not expect hyperinflation so to the extent their forecasts are accurate the recent surge in US spending as response to the recent crisis as well as the increases in money supply seem to not be large enough to cause it yet.
The relationship between money printing and inflation is still debated among economists (Austrian, Keynesian).
One framework that is relevant to your question is the ratio of productivity to money supply. In a simple scenario, if more money is chasing the same quantity of goods and services, then inflation can be reasonably expected (ceteris paribus). The optimistic note here is that QE has conventionally only brought inflation to a narrow group of asset classes, not the broader economy. However, the bad news is that the Fed in 2020 is expanding its scope of asset purchases and in conjunction with fiscal "helicopter money" there may be some cause for concern.
This is one of the more debated topics, but it's empirical that there is at least a correlation between low rates/QE and low productivity -- which does not bode well for long run inflation. But again, in the long run we're all dead.
Scope of purchases
A good example is in 2008, the narrow asset classes of QE purchases limited inflation to the financial economy. As a result, a US citizen, evaluated at the median, with exposure to the real economy did not see much inflation in CPI or PPI.
Inflationary vs deflationary
In an economy where austerity is setting in (quarantine, layoffs, ect), there is a deflationary force spreading at the macro level. The act of printing money can offset this; thus, the result would not be inflation, but rather maintaining initial conditions. Of course, it is extremely difficult for policy makers to strike the right balance. Often times policy makers prefer aggressive stimulus than austerity, because it's less painful (lesser debt burden on society).
Historical cases, like Argentina as you mentioned, are frequently triggered by debt burdens out of the central bank's control; foreign denominated debt can't be printed by the nations central bank. (i.e. Greece, Weimar Republic Germany). The US is uniquely positioned as having the world's reserve currency and it would take gross negligence/malfeasance from policy makers to allow for hyperinflation with so many policy levers at their disposal. Nonetheless, policy impotence is still a reoccurring factor and should not be ruled out.
Hope this helps you think of it in a few new ways.