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> I can't think of many cases when saying "This algorithm is O(N^2)" is significantly more useful than saying "This algorithm is going to be really slow".

But to know that it will be slow you have to know how it scales with the input size. Because the list example may work just fine when the website has little data (oftentimes an O(N^2)-algorithm may even be faster than an O(N)-algorithm on small data) and hit you when you scale up. And the O-notation gives you a tool to estimate how the work scales. Any alternative to the O-notation is either more complex (look up Analytic Combinatorics http://ac.cs.princeton.edu/home/) or just an informal treatment of O-notation.

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