You never quite state the alternative to an algorithm. I propose that the only alternative is randomness.

The alternative to algorithms are non-formalizable processes. Obviously I can't give a precise definition or example of one, since in this case we would have an algorithm again.

The best example I can give is the following: Assume that the universe works precisely according to laws (I don't think so, but let's assume it). What determines the laws? Another law? If so, you get an infinite regress of laws, and you don't have a law to determine this, either. So according to you, the laws of the universe are random. I think this hardly plausible.

I'm curious as to what you think the human brain does, if not an algorithm.

I don't know, and I don't think it is knowable in a formalizable way. I consider intelligence to be irreducible. The order of the brain can only be seen in recognizing its order, not in reducing it to any formal principle.

In addition, I think the human brain is a lot more algorithmic than you think it is. A lot of Lukeprog's writings on neuroscience and psychology demonstrate ways in which our natural thoughts or intuitions are quite predictable.

I am not saying the human brain is entirely non-algorithmic. Indeed, since the knwon laws of nature we discovered are quite algorithmic (except for quantum indeterminateness) and the behaviour of the brain can't deviate from that to a very large degree (otherwise we would have recognized it already) we can assume the behaviour of our brains can be quite closely approximated by laws. Still, this doesn't mean there isn't a very crucial non lawful behaviour inherent to it.

The universe started with the laws of physics (which are known to be algorithms possibly with a random generator), and have run that single algorithm up to the present day.

How did the universe find that algorithm? Also, the fact that the behaviour of physics is nicely approximated by laws doesn't mean that these laws are absolute or unchanging.

What do you think about my proposed algorithm/random dichotomy?

Frankly, I see no reason at all to think it is valid.