Who launches a private non-profit university? Universities are normally non-profit. For public universities, it is understandable that governments invest the taxpayers' money for education purpose.

However, investors of a private university should invest a huge amount of money (campus, buildings, facilities, staff, etc.) without ambitious hope for financial gain. At best (after a few years), a new university can earn enough money (tuition fee and donations) to operate without financial benefits for its original investors.

Who invests in private universities? 

It cannot be financial investment, and not all cases are charity. Then, how does this system work? as it is rapidly growing in many countries.

Who makes the huge initial investment?
I'm only familiar with the US, but my understanding is that most (if not all) private non-profit universities were started with a single large gift from a wealthy individual, family, small group of individual, or a charitable foundation (usually started by one of the above).  In many cases the institution is named after that person.

I'm not sure in which countries the number of private non-profit universities is "rapidly growing".  I only know of a few recently started institutions in the US.  One with which I'm somewhat familiar is Olin College.  According to their web site, the college was started with a $460 million gift from the F. W. Olin Foundation, which in turn was founded in 1938 by Franklin W. Olin and funded by his personal wealth.

Such gifts are considered "charity" in that their main purpose is to benefit the community, and the donor doesn't receive a financial benefit.  Of course, any given donor could certainly have personal motivations other than a genuine desire to do good (attention, guilt, fun, taxes, etc.), but there is really no way to guess.