EDITED (I realized that the question certainly needs a context)
The problem 1.1-5 in the book of Thomas Cormen et al Introduction to algorithms is: "Come up with a real-world problem in which only the best solution will do. Then come up with one in which a solution that is “approximately” the best is good enough." I'm interested in its first statement. And (from my understanding) it is asked to name a real-world problem where only the exact solution will work as opposed to a real-world problem where good-enough solution will be ok.
So what is the difference between the exact and good enough solution. Consider some physics problem for example the simulation of the fulid flow in the permeable medium. To make this simulation happen some simplyfing assumptions have to be made when deriving a mathematical model. Otherwise the model becomes at least complex and unsolvable. Virtually any particle in the universe has its influence on the fluid flow. But not all particles are equal. Those that form the permeable medium are much more influental than the ones located light years away.
Then when the mathematical model needs to be solved an exact solution can rarely be found unless the mathematical model is simple enough (wich probably means the model isn't close to reality). We take an approximate numerical method and after hours of coding and days of verification come up with the program or algorithm which is a solution. And if the model and an algorithm give results close to a real problem by some degree that is good enough soultion.
Its worth noting the difference between exact solution algorithm and exact computation result. When considering real-world problems and real-world computation machines I believe all physical problems solutions where any calculations are taken can not be exact because universal physical constants are represented approximately in the computer. Any numbers are represented with the limited precision, at least limited by amount of memory available to computing machine.
I can imagine plenty of problems where good-enough, good to some degree solution will work, like train scheduling, automated trading, satellite orbit calculation, health care expert systems. In that cases exact solutions can't be derived due to constraints on computation time, limitations in computer memory or due to the nature of problems.
I googled this question and like what this guy suggests: there're kinds of mathematical problems that need exact solutions (little note here: because the question is taken from the book "Introduction to algorithms" the term "solution" means an algorithm or a program, which in this case gives exact answer on each input). But that's probably more of theoretical interest. So I would like to narrow down the question to:
What are the real-world practical problems where only the best (exact) solution algorithm or program will do (but not the good-enough solution)?
There are problems like breaking of cryptographic ciphers where only exact solution matters in practice and again in practice the process of deciphering without knowing a secret should take reasonable amount of time. Returning to the original question this is the problem where good-enough (fast-enough) solution will do. There's no practical need in instant crack though it's desired. So the quality of "best" can be understood in any sense: exact, fastest, requiring least memory, having minimal possible network traffic etc.
And still I want this question to be theoretical if possible. In a sense that there may be example of computer X that has limited resource R of amount Y where the best solution to problem P is the one that takes not more than available Y for inputs of size N. But that's the problem of finding the particular solution for P on computer X which is... well, good enough.
My final thought that we live in a world where it is required from programming solutions to practical purposes to be good enough. In rare cases really very very good but still not the best ones. Isn't it? :) If it's not can you provide an example? Or can you name any such unsolved problem of practical interest?