How many crossing changes does it take to get to a homotopy trivial link?

An intensely studied problem in knot theory is to determine the unlinking number of a link; that is, the number of crossing changes needed to transform a link into the unlink. We ask a related question: How many crossing changes are needed to turn a link into a homotopy trivial link? A link is called homotopy trivial if it can be transformed into the trivial link by allowing components to pass through themselves but not each other. We determine this homotopy trivializing number for all 4-component links up to link homotopy. In particular, we provide a table where one can read off the homotopy trivializing number of a link given its linking data (pairwise linking number and higher order Milnor invariants). Moreover, we give upper and lower bounds on the homotopy trivializing number for links with an arbitrary number of components, showing that the difference between the homotopy trivializing number and the sum of the pairwise linking numbers is bounded by quadratics on the number of components.