From 2008 to the present, the Federal Reserve has created large quantities of reserves: has "printed" a lot of money: more than doubling the supply. We know that in the long run, inflation moves 1:1 with money. Why hasn't there been inflation?
Note that Federal Reserve liabilities have increased dramatically: liabilities rose dramatically due to emergency lending measures. These measures were temporary, but the reserves were reinvested into other assets. Using the Federal Reserve's H.41 historical release plus discontinued series, Corrections displays the Federal Reserve's assets, in a graph inspired by James Hamilton (click to enlarge).
Rather than create money and then retire it, the Federal Reserve reinvested money created for loans into mortgage backed securities and treasury bonds. This is using printed money to finance a deficit, no doubt.
So why no inflation? One answer, losing credibility for the last five years and counting, is that prices just haven't had enough time to react. A better answer is that unlike a money drop, the Federal Reserve printed money and put it into the economy, purchasing assets. Those assets slowly mature, and take money back out of the economy, allowing the Fed to retire reserves.
Imagine the Fed creates $1 and purchases a Mortgage-Backed Security that matures 10 years from now selling at a 50% discount. For 10 years, the economy will have one more dollar, and one less MBS in circulation. Come the end of those ten years, the MBS pays back $2, which the Fed takes to retire reserves, having originally pressed a button to create only $1. Then the economy now has one less dollar in reserves than before all this, and the Federal Reserve has one more.
There is no magic in all this: the point is only to emphasize that the Fed isn't just printing money, creating a liability. It creates liabilities and assets, generally at a one-to-more-than-one ratio in the long run. This allows it to have low inflationary effect in the long run, because in the long run, no new reserves are created.
Corrections displays a very simple example of how money creation to purchase assets is inflation neutral (click to enlarge): anyone can do what the Fed did, in theory: go into debt to purchase assets, "freeing" money that other people can spend (the person who sold the asset now has a credit in their bank account equal to your debit). Use the returns for those assets to retire your debt, and the economy is back to where it was: nothing new is created in the end.
Will the Fed retire those reserves? Skeptics will say that historically, governments don't do that. They're wrong. Throughout its history Britain would go to war, and cease convertibility to gold. It would "print" money (go into debt). Then, in the years after the war, it would run surpluses and reduce monetary aggregates until the gold standard could be resumed at the same level it was previously. For example, from 1797 to 1821 Britain departed the gold standard during the Napoleonic Wars, returning to where it left after 24 years. This happened again in the U.S. after the Civil War, as the U.S. returned to prewar parity. This happened yet in Britain after WWI in 1925, as they resumed at prewar parity.