As an example, POTUS does not undertake the entire pardon process on his own, instead they are preliminarily processed by the Department of Justice, which makes a recommendation. Only POTUS can "grant" the pardon, and we lack substantive information on how deeply he gets involved (does he just sign documents based on staff recommendations? does he do his own research?). I don't know what physical form Clinton's 450 pardons on January 20, 2001 took, but a president might use an autopen (Obama did use such a device). Questions have been raised and definitive answers from SCOTUS are lacking as to whether hand-to-paper signatures are demanded by the Constitution. DoJ at one point thought it was okay.
This article lists the Constitutional functions which cannot be delegated:
reporting to the Congress on the state of the Union; convening either
or both Houses of Congress and adjourning Congress; signing and
vetoing bills; receiving ambassadors and public ministers
(recognition); appointing and removing ambassadors, ministers, and
certain other public officers; nominating federal judges; and making
recess appointments
However, the Constitutional function of negotiating treaties or numerous functions as commander in chief of the armed forces is generally delegated, within the executive branch. Numerous statutory functions can be delegated (and Congress may say explicitly that some function can be delegated, in passing the law). The matter of implied power to delegate statutory function is the topic of a couple of centuries of litigation, but as an example in Williams v. United States, 42 U.S. 290 SCOTUS held that a law
prohibiting the advance of public money in any case whatsoever to the
disbursing officers of government except under the special direction
of the President does not require the personal and ministerial
performance of this duty, to be exercised in every instance by the
President under his own hand.
SCOTUS has not established bright lines regarding the extent of permissible delegation.
