Billionaire Michael Bloomberg’s essentially limitless budget for his Democratic presidential bid adds uncertainty to an already crowded and unsettled race while increasing the odds of a prolonged nominating battle in a contest that has yet to produce a definitive front-runner.

The 77-year-old former New York City mayor announced his bid on Sunday after indicating earlier this month he might enter the race. Mr. Bloomberg faces a tough path to the nomination given his late entry and lack of a campaign apparatus, though he will...