"Functional" may be a bit of a misnomer, but we already know that the vast majority of DNA is not used to encoding proteins. The rest of the DNA contains information for things like miRNAs which are extremely important for gene regulation, but again, these regions only serve as a regulatory mechanism and not to greatly expand the amount of proteins possible.
We also now know that humans have less protein enconding genes than some strains of rice, yet who'd argue that rice is more complex than a human based on the sheer size of an encoding genome? The obvious argument people will try is that genes can be spliced in multiple ways to make different proteins. Well, the entire proteome, that is the grand total of all proteins , which of course reflects all of the entire output of the genome, is only about ~100K-200k proteins last time I checked, which can still be argued as still far too small to contain the complexity that defines a human. What people have ignored for a long time , however, have been the posttranslational modifications (PTMs) that get added to proteins after they're made. There are over 400 different known types of these modifications. Glycosylation of proteins is the largest class of PTMs. If you sum the total number of glycan structures that get added to proteins, you get what has been dubbed the 'glycome', which is akin to the genome for DNA and then proteome for proteins. The glycome is now known to be orders of magnitude more complex than the entire genome and proteome and in fact has been described as one of the most complex entities in all of nature. Carbohydrate structures on proteins can endow them with entirely new functions and can even supercede the importance of the protein itself for function. Unlike DNA and proteins , however, you can not control PTMs in a template like manner because there is no code that exists for controlling glycosylation like there is for DNA. The 20th century was dominated by DNA and genetics, but I'll put all of my money on the fact that the next revolution in biology and most of the effort in the next two centuries will be spent on deconvoluting and trying to figure out how to tame the realm of PTMs. PTMs are where the real molecular diversity occurs that produces the millions or even billions of distinct molecular species at any given moment in time that truly define life. Life is certainly sweet, and it is why there has been an explosion in carbohydrate biology research and why glycobiology has been named as one of the top 10 fields of science that will transform medicine in the 21st century by MIT. The trillions of possible combinations through PTMs is how such a small protein coding genome and small proteome can be transformed into a grand set of molecular complexity that is needed to create a human.