For many years, Waseda received the most applications for its undergraduate program. Recently, 明治大学 has surpassed it (over 113,000 applications in 2011, with only 1 out of nearly 25 students being offered admission), mainly based on its decision to open up as many application channels as possible. That means that prospective students can apply on the basis of recommendations, the センター試験, and many other categories.
I can only assume that this is a strategy to stave off the decline in student numbers that private universities are facing throughout Japan based on demographic developments and their dependence on tuition payments for funding.
Another case further down the prestige hierarchy is Toyo University (東洋大学) which has parlayed success in the (often televised) Ekiden (駅伝) long-distance races into national prominence and an increase in application.