The act of 'measuring', limiting or slowing down is something that the philosopher Gilles Deleuze has written about quite extensively. He seems to have inherited this particular concept from Leibniz, who is in some ways the inventor of the limit (which, as you correctly note, is the ground for the possiblity of measurement), and who also wrote quite a bit on the subject. It also takes on a Kantian aspect in Deleuze (like most of his concepts), whose Critique of Pure Reason (in particular the chapter 'Transcendental Aesthetic') can also be seen as a meditation on the idea of measurement (for Kant, space [and therefore measurement] is no longer a relation [e.g. between bodies] as it is in Leibniz, but the form of appearances itself).
According to Deleuze, measuring (although its not a term he uses himself, nor does it appear to have any significance in English translations of his work) is the defining gesture of science and scientific thought, and is related to the primary functives of science (functives are the things which a scientific function is made of, and functions are the objects of science):
The first functives are therefore the limit and the variable, and reference is a relationship between values of the variable or, more profoundly, the relationship of the variable, as abscissa of speeds, with the limit (What is Philosophy?: 'Functives and Concepts')
For Deleuze, the universe consists of infinitely many forms appearing and dissappearing at infinite speeds ('chaos' or 'chaosmos'). Science uses measurement to help us deal with the utterly incomprehensible chaos (hence the 'Kantian aspec') by slowing it down:
It is these first limits thaat constitute a slowing down in the chaos or the threshold of suspension of the infinite, which serve as endoreference and carry out a counting: they are not relations but numbers, and the entire theory of functions depends on numbers. We refer to the speed of light, absolute zero, the quantum of action, the Big Bang: the absolute zero of temperature is minus 273.15 degrees Centigrade, the speed of light, 299,796 kilometers per second, where lengths contract to zero and clocks stop. Such limits do not apply through the empirical value that they take on solely within systems of coordinates, they act primarily as the condition of primordial slowing down that, in relation to infinity, extends over the whole scale of corresponding speeds, over their conditioned accelerations or slowing-downs.
So to answer your question, Deleuze might say that the requirement for measuring (and he would be pleased that this question is indeed a Kantian question) is precisely the limit:
Yet it is not the limited thing that sets a limit to the infinite [thereby allowing for the possiblity of measurement] but the limit itself that makes possible a limited thing. Pythagoras, Anaximander, and Plato understood this: the limit and the inifite clasped together in an embrace from which things will come.