Term for when the more you invest yourself in something, the less you agree to drop it
Not certain this is what you are thinking about, but this sounds a lot like the idea of "sunk costs", which is a form of loss aversion.

Sunk costs means that you tend to overvalue the effort you have already put in to something (time, money, etc), and often are willing to put in more effort to try to rescue what you already put in, even though the additional investment (again: time, money, etc) is not worth it. Often people make poor decisions (financial, relationship, etc) because they feel like they have "put too much in" to it.

A related concept is the Endowment effect (first mentioned in another answer by JoaoBotelho). Like sunk costs, some of the endowment effect can be explained by loss aversion, though some experiments have also shown a smaller mere ownership effect which is not also explained by loss aversion. These effects describe how people tend to put more value on things they own or have some relationship to (for example, an object held versus an object at a distance) versus things they do not.