Y Combinator has transformed itself from an experiment in angel investing into a Silicon Valley kingmaker.

As Dropbox Inc. begins trading as a public company, it marks the first IPO by a Y Combinator company and a milestone for the accelerator behind other hit companies like Airbnb and Reddit. Although Y Combinator still nurtures scores of startups like those stars, it has busied itself with a wave of growth initiatives like raising capital from endowments, pushing harder into later-stage investing and backing more enterprise...