An AnCap society probably won't have corporations with tens of thousands of employees, but it also will lack the institutional reasons such megacorporations exist. Transactions between business A and business B in today's world require a considerable amount of book-keeping and tax overhead. Each entity must record details of the transaction, over and above those needed for their own business purposes, to satisfy the tax lords. Taxes must be paid, or in the case of tax-exempt transactions, proof of exemption must be maintained. This encourages entities to integrate vertically, over and above any innate business advantage of doing so. Regulatory overhead, in a large corporation, can be amortized over many transactions. A large corporation may have entire teams who deal with regulatory compliance alone.
Odds are that corporations will tend to be smaller. How much smaller, we can only guess.
I suspect that an AnCap society will have a lot more reserve capacity than is available today. Supporting $3.5 trillion of federal government, plus an approximately equal amount of state and local government spending, is a huge drain on productivity. Some people will choose to work as hard as before, which will allow them to produce much more than before. Some will choose to spend a lot more time to "stop and smell the roses"; some will engage in more long-term prospective research and intellectual activity.
Having such great reserves of time, savings, and productive capacity, will mean the potential for vast expenditures on whatever purposes may inspire people, or voluntary associations of people - including education, health care, defense, space travel, and so forth.
How inefficient is government? Let's take education, which swallows about half of all local and state spending. Home schoolers are able to do much better with less than a third of the time used by government schools, and vastly less expense. A very conservative estimate would be that a better education could be accomplished for a tenth of the total resources now devoted to that field. Thomas Jefferson, based upon the usual level of attainment in his day, expected that a child with just three years of formal education would be able to keep books for his or her own business, and read, understand, and amend legal contracts. Frankly, he underestimated the capacity of children to learn - but these attainments seem remarkable today, so used are we to the lousy performance of government schools.