How many ways can a major league baseball player make it to first base without a hit?

According to this article at SBAR - Society for American Baseball Research - there are six ways to get to first base without a hit: walk, error, fielders' choice, hit by pitch, dropped third strike, and defensive interference. The article was published in the Spring 2014 edition of the "Baseball Research Journal". The author, Paul Hertz, list six games in which each has occurred.

https://sabr.org/journal/article/was-there-a-seven-way-game-seven-ways-of-reaching-first-base

According to this page, depending on how one wants to separate errors or types of walks, there are as many as 8 ways to reach first without hitting the ball:

https://www.theoleballgame.com/how-many-ways-to-reach-first-base-without-a-hit.html

The second link just breaks down two of those given in the first into different ways that each can happen. However, as the question mentions "player" and not "batter", another way of reaching first base is as a pinch runner.

The second link just breaks down two of those given in the first into different ways that each can happen. However, as the question mentions "player" and not "batter", another way of reaching first base is as a pinch runner.

A pinch runner is a substitute used for a runner who is already on base.
https://www.baseball-reference.com/bullpen/Pinch_runner

A pinch runner is a player substitution, the person at bat would have to successfully reach base before the substitution.

Official Playing Rule 5.06(c)(7) Comment (last paragraph).

If a pitched ball lodges in the umpire's or catcher's mask or paraphernalia, and remains out of play, on the third strike or fourth ball, then the batter is entitled to first base and all runners advance one base. If the count on the batter is less than three balls, runners advance one base.

https://baseballrulesacademy.com/special-report-ball-sticks-catchers-protector/