Or really a minimal version of any genome. Build me a fruit fly with the "non-coding" sequences removed, and show me that its phenotypically normal. The genomes are remarkably similar, thus proving the point while not involving potential ethical issues.

Note, that I don't think this would be an easy experiment. For example, if you removed all the Alus, you would likely break something critically functional, but not be able to conclude specifically that it was due to removing the Alu, rather than, say, some other complicated genomic effect.

Another way to look at it: pick a model organism (maybe c elegans), run many experiments in parallel, finding regions which can be removed and allow viability under a wide range of inducible environments (if you remove the heat shock proteins, but never put the worms in a hot environment, you might conclude you increased fitness). Find the largest regions you can remove independently, then start with pairs of regions (you can imagine that removing region 1 and region 2 independently would have no effect, but be fatal if combined), etc.

The resulting genomes would probably still contain regions that contained no recognizable function, have no strong selection pressure, and yet still be completely necessary for viability. This is just speculation on my part, though.