Roads, irrigation, education, research, and safe water. There is reason to suspect that, left alone, markets will systematically undersupply these goods. One reason is the public nature of these goods - in economics, we use the terms "non-rivalry" and "non-excludability". The first denotes a condition in which my consumption doesn't reduce yours - think of the act of reading this weblog entry (doesn't prevent others from reading it too; however, my site's bandwidth and your ISP connection is a rival good.) The second denotes high cost of preventing others from using the good. For example, while large high-speed roads can be made into tollways, rural roads cannot be feasibly made into tollways! Related to this are "externalities" - outcomes that inadvertently spill over to others - associated with these goods. Safe water prevents the spread of contagious diseases (the "contagious" should clue you in on the externality.) Education supposedly makes people nicer, more respectful of social order, and more informed about political and social options - crucial when decisions have to be made collectively, like choosing the President.
Knowledge is a public good. It's non-rival; excludability is often difficult (but not impossible for some types of knowledge - remember Coke's famous trade secret?) So R & D, which produces scientific knowledge, will be undersupplied by the private sector.
Pop quiz: government intervention in agriculture often involves supply of "postharvest facilities" - mechanical driers, storage centers, etc. - as well as "price stability" - the government engaged in commodity trade to reduce price volatility (as exemplified by the Philippines' National Food Authority). Can you explain why postharvest facilities and price stability will be systematically undersupplied by the market?