What is the largest infiltration of the US government we know of?

I guess my first question is: what do you mean by "infiltration"?

If you mean in terms of espionage, I'd say it's probably the Soviet infiltration of the federal government during FDR's administration. During the late 1930s and all the way through the Second World War, the NKVD (predecessor to the KGB) had agents in place at the highest levels of the US government.

These are just some of the Soviet agents who worked for the NKVD immediately prior to and during the Second World War:

Alger Hiss: a senior State Department official, and one of the leading US officials involved in the establishment of the United Nations.
Lauchlin Currie: an economic policy adviser to President Roosevelt;
Duncan Chaplin Lee: a US army lieutenant-colonel and personal aide to Bill Donovan, the founding director of the CIA predecessor agency OSS.
Harry Dexter White: Assistant Secretary of the Treasury and the lead US negotiator at the Bretton-Woods conference.
Lawrence Duggan: another senior official in the State Department, and head of the Latin America desk.

Those last two are particularly interesting in one plausible counterfactual — intelligence historian Christopher Andrew writes: "Henry Wallace, vice-president during Roosevelt's third term of office [...] said later that if the ailing Roosevelt had died during that period and he had become president, it had been his intention to make Duggan his Secretary of State and White his Secretary of the Treasury."

In fact, Andrew argues that this "breathtaking gulf" in the intelligence capabilities of the US and USSR gave Stalin "a considerable negotiating advantage" at the crucial Tehran Conference (one of the meetings of the 'Big Three' — Stalin, Roosevelt and Churchill) in 1943 — made worse by American naivete about espionage:

[FDR] accepted Stalin's pressing invitation that — allegedly on security grounds — he should stay at a building in the Soviet embassy compound rather than at the US legation. It seems not to have occurred to Roosevelt that the building was, inevitably, bugged, and that every word uttered by himself and his delegation would be recorded, transcribed and regularly reported to Stalin.

The situation was even worse at the Yalta conference in 1945:

Yalta was to prove an even bigger success for Soviet intelligence than Tehran. This time both the British and the American delegations, housed respectively in the ornate Vorontsov and Livadia Palaces, were successfully bugged.

This is during a period in which neither British nor American intelligence had any active agents in Moscow — and also the period in which the US had only just established its first professional intelligence agency (the OSS was established in June 1942.) Right up until the late 1940s and early 1950s, when the Venona decrypts enabled the US and UK to wind up much of the Soviet's spy network, the NKVD had eyes and ears in many of the most sensitive corners of the US government.