How does the Objectivist politics limit the power and influence of corporations?

Outside of regular and perfectly rational concern for objectivity in pursuit of justice, if politics has no power to influence the economy then there is no reason for economic entities to influence politics for economic purposes.

The real issue is: how do we keep politics from interfering in economics in the first place? The answer is laissez-faire, and as a necessary philosophic foundation and practical prerequisite for that, a population constituted predominantly of individuals who understand in at least some form the epistemological and moral basis for laissez-faire and who act accordingly anywhere that thought about, discussion of and action upon the issues are indicated (eg inter-personal discussions, school as appropriate, the marketplace, the voting booth, and so on).

JJM

Corporations would be subject to the same limitation that everyone else is: they cannot violate individuals' rights through force or fraud.

The short answer:

There is only one standard of justice in the field of economics: the verdict of a free market. - The Ayn Rand Letter

What power and influence to corporations have? Doesn't it mostly involve the power to influence government to make laws that favor their business or disfavor their competitors? The first step in protecting against that is to not allow government to make such laws. There should be a clear separation of economics and state, in the same way and for the same reasons as the separation between church and state.

With government influence in the economy removed, any remaining potentially bad power and influence would involve the use or threat of force or fraud -- which are crimes. Protecting individuals from that kind of thing is the only valid role of government.

All that remains is the ability for corporations to try to persuade others, peacefully, using reason. There's no need to curtail that type of power or influence; it's something that everyone participating in an economy has, to one degree or another.

There is a widespread misconception that if left to their own devices corporations would expand and grow and take over the world. Presumably this is the unstated aspect of the questioner's question that has not been addressed by previous answers.

In microeconomics (i.e. the economics of a corporation or a household) there is a powerful concept of "economies of scale." The most famous example of this is Henry Ford's ability to drive the price of cars down so low because he was able to mass produce them.

While economies of scale is hugely important, an equally important concept (which is not discussed nearly as much as it should be) is DISeconomies of scale. The larger the organization the more "friction" there is to getting things done. Talk to anyone who's worked in a fortune 100 company and you'll quickly get overwhelmed by stories of inefficiencies and problems that hold the company back. Stories of political infighting, factions waging war against other factions, product A in the company stealing market share from product B in the company, misappropriation of funds, cronyism, etc. All of these limit the growth of a company.

Note, in the above I haven't even mentioned competition, which is another external force that dooms all companies from getting too large. Thus, the entire concept of a company getting "too large" or "too powerful" is an invalid one in a free economy. If the government starts to dictate favors as many of the other posters have pointed out, then the companies can get larger, but these forces will still limit their growth.

The laws of economics are just that. Laws. They cannot be violated. They are as immutable as the laws of physics.