What three institutions influence economics the most?
I think that you could get a better answer to this question if you could be more clear about what you are asking. When you mention "institutions" what are some examples of what you want discussed? When you say "economics" do you mean the academic discipline of economics or do you mean the economy of a given country?
With regard to the economy of a given country, the "institution" that seems to have the most influence on that is the government. The level of influence that the government has varies from country to country, but it is strong even in relatively laissez faire places like the US. Here in the US, for example, there are even rules (in my state) mandating that children in any day care setting must have personal folders in which they can put things that they want to show their parents.
With regard to the discipline of economics, I would say that the "institution" of academia has the most influence. Professors of economics are the ones who come up with new theories that affect the discipline.
Perhaps both the economy and economics are influenced by the "institution" of private think tanks. These entities can have researchers who suggest new ideas, but they can also have an impact on the economy by pushing for new policies or by supporting new technologies.