What was the federal government's involvement in the United States economy from 1900-1945?

This is a relatively long time period that you are discussing in this question. The federal involvement in the economy ebbed and flowed during this period. The early part of the Twentieth Century, until the American involvement in the Great War, was a period of progressive reform. The government was very active during this period in terms of regulation. Efforts were made to regulate monopolies and workers were granted unprecedented rights. These rights included the right to collective bargain and form unions. Banking reforms, including the Federal Reserve Act, were also made during the Progressive Era.

As the United States entered World War I, the federal government took drastic steps to regulate the economy and mobilize it for war. The government went as far as to fix prices, wages, and even told certain companies what to produce. After the war, the federal government was controlled by Republican presidents during the 1920's and took a laissez-faire approach to the economy. Private companies enjoyed a period of autonomy that had not existed in the Twentieth Century. The stock market crash of 1929 would end the era of economic conservativism.

Franklin D. Roosevelt's overwhelming victory in the 1932 election marked a key turning point in the role of the federal government. Because of the drastic economic depression, the American public wanted the government to act. Roosevelt responded with a program of federal intervention on the economy that was not seen to that point in history. The National Recovery Act fixed prices and wages but was eventually declared unconstitutional by the Supreme Court. The New Deal, as it was called, also made the federal government a major employer by offering major public works programs that hired hundreds of thousands of Americans. The New Deal also regulated banks and corporations in a major way. During World War II, similar mobilization efforts were made to prepare the nation for war.  It could be considered a striking success as the United States economy was fixed by the war effort. While there are still ebbs and flows with regards to federal government's role on the economy, the New Deal was a turning point in this area.