How much money can I bring into the Philippines by plane?

There is no limit as to how much money you can bring in.

There is only a requirement that you declare any amount that is over PhP 400,000 equivalent or $10,000 USD. You do this on the Customs form when arriving at the airport. This is an international agreement that was put in place in 2001 to limit money laundering. It will not hinder you from bringing money into the Philippines, as it is only a report.

IMPORTANT: If reporting over $10k or more, make sure you state "personal income" as the source of income on the Customs Form, or there will be further regulations.

US Citizens

If you are a US Citizen, there are US reporting requirements for bringing money out as well. Here's an excerpt from
U. S. Customs:

However, if a person or persons traveling together and filing a joint declaration (CBP Form 6059-B) have $10,000 or more in currency or negotiable monetary instruments, they must fill out a "Report of International Transportation of Currency and Monetary Instruments" FinCEN 105 (former CF 4790).

Note, that it is per persons traveling together (not per person in the family). A family must all be reported as traveling together.