Pharmaceutical companies are ringing in the new year by raising the price of hundreds of drugs, with Allergan PLC setting the pace with increases of nearly 10% on more than two dozen products, according to a new analysis.

Many companies’ increases are relatively modest this year, amid growing public and political pressure on the industry over prices. Yet a few are particularly high, including on some generics, the cheaper alternative to branded accounting for nine out of 10 prescriptions filled in the U.S. Overall, price increases,...