Sure! Most real numbers are uncomputable, for starters. "approximations are usually good enough" is a smoke screen that assumes that it's always possible to make approximations and iteratively improve their accuracy, something which generally is only true for computable numbers!

Another example, something I've been studying recently, is the phenomenon of patterns. Patterns are very abstract, and it's not at all clear how a computer can model a pattern without resorting to some concrete instantiation of it. At best, we have blueprints, models, code, designs, and instances; these are all themselves occurrences of patterns, but they fail to embody the pattern itself.

While I've got your attention, "Cryptography also enables a high level of security." is a ridiculous line. Cryptography, by itself, is only a building block. Security is structural. Check out object capabilities: http://srl.cs.jhu.edu/pubs/SRL2003-02.pdf

Edit: Scott Aaronson discusses limits of quantum computation in this talk: http://www.scottaaronson.com/blog/?p=2903 Basically it is totally possible that minds cannot be simulated by computers without destroying the mind in question!