Stanford President John L. Hennessy said the Palo Alto, Calif., school will gradually expand its entering classes until it reaches a comfort level with its student population. The university now has about 7,000 undergraduates. (Paul Sakuma/AP)

The most selective university in the country plans to open its gates a bit wider in the next few years.

Stanford University, which turns down roughly 19 out of every 20 applicants, wants to grow its entering freshman class by an estimated 100 students in the fall of 2016. That would translate to a class of about 1,800.

Stanford President John L. Hennessy said the private university in California’s Silicon Valley will in ensuing years gradually expand its entering classes until it reaches a comfort level with its overall student population. The university now has about 7,000 undergraduates. How many more it can or will accommodate is an open question.

“We don’t know at what point does the size diminish the experience,” Hennessy told The Washington Post during a recent trip to Washington. “Clearly it does at some point. But we don’t know if that’s at 8,000 students, or 9,000, or 10,000. We just don’t have any idea.

“But we like the notion that the undergraduate body’s at a size where many of the students know many of their fellow students, where they can have a shared experience that’s highly valuable. We don’t want to lose that in the process.”

He said the university will expand dormitories and faculty and then gauge feedback from students and professors in coming years to determine how big is too big.

Stanford is not the only ultra-selective school with expansion plans. Yale University announced in June that it had raised enough money to build two new residential colleges by 2017, enabling undergraduate enrollment to grow from 5,400 to 6,200.

Many of the biggest names in private higher education have grown little in the past decade even as demand has soared. Few of the nation’s elite liberal arts colleges, which promise small class sizes and robust residential education, have appeared interested in growth.

Federal data shows that the entering class at Williams College — which U.S. News and World Report calls the top liberal arts school — numbered 550 in 2009. In following years the totals — 546, 548, 545 and 547 — were utterly stable. This pattern also held for colleges such as Swarthmore, Carleton, Amherst, Pomona and Vassar.

In the top tier of private research universities, the data shows measurable expansion in the entering classes at Rice University (from 789 in 2008 to 978 in 2013), Washington University in St. Louis (from 1,429 to 1,610 in the same span) and the University of Chicago (from 1,307 to 1,426). But otherwise, growth has been scant among this group, too.

To some extent, prestigious schools have little incentive to grow. If undergraduate enrollment remains stable while applications rise, then admission rates fall. Lower admission rates help schools maintain their status in the U.S. News rankings and other subjective measures, because the schools are considered more selective.

But there are potential upsides to growth. More students could mean more tuition revenue, depending on the financial need among those admitted and how much aid they receive. More alumni means more global exposure and a broader base of potential donors.

There is also an argument that it is the right thing to do.

“The elite private universities are essentially the same size they were 40 years ago,” Hennessy told an alumni magazine last year. “Meanwhile, the demand among highly prepared students has grown quickly. Public institutions have taken up most of the slack but that was in a time when the publics were healthier than they are today. A handful of private schools are not going to solve the problem, but don’t we have some moral responsibility to step up and educate a greater fraction of these highly talented students?”

During the previous admission cycle, 42,000 students applied to Stanford. Barely 5 percent were admitted, the lowest rate in the country. Hennessy acknowledged that expanding the incoming class won’t necessarily raise the rate if application numbers continue to rise.

Asked if the low admission rate bothers him, Hennessy told The Post that it is never easy to turn down so many highly qualified applicants. “You make a small number of people happy, and you make a number of really talented students and their families unhappy,” he said. “I don’t know what to do about that.”