“All of those cases are red meat to a lot of people,” says Dan Ortiz, a law professor at the University of Virginia. “And there are things in the pipeline that could make things even more interesting.”

In terms of public visibility and legal impact, the 2019 docket could eclipse anything the court has handled since Trump appointees Neil Gorsuch and Brett Kavanaugh joined. Some of the decisions are likely to land near the term’s end in June. That timing could thrust the court into the political arena just as the 2020 presidential contest comes into focus, particularly if the justices step in to resolve subpoena battles or related power struggles between Mr. Trump and congressional Democrats who are investigating him.

At the center sits Chief Justice Roberts, a thoroughly conservative jurist who heads perhaps the most conservative Supreme Court in 80 years. He is dealing with competing pressures: seizing the opportunity to implement the rightward legal vision that animated his career versus exercising the restraint many believe necessary to preserve the court’s credibility. The chief justice also is facing the prospect of presiding over any Senate impeachment trial of Mr. Trump.

It is a “misperception” to view the court as a political body, Chief Justice Roberts said last month in New York. Of 19 cases resolved by 5-4 votes last term, only seven fell along perceived ideological lines, he noted. “It shouldn’t come as a surprise because we don’t go about our work in a political manner,” he said.