Three guys reached 190 since 2004: Pujols with 192 in 2008, Miguel Cabrerra with 190 in 2013 and Harper with 198 in 2015.
His path might be easy. The Angels have only 3 games left against any of the top 4 teams in the AL in ERA+ (Bos, Clev, NY, KC). That team is Clev. Here is the OPS+ of those teams
The next highest ERA+ the Angels will face is the 102 of Texas & Tampa Bay and Texas traded Darvish. The Angels have 15 games left against 3 of the 4 lowest ERA+ teams in the AL.
Bal 89 (5)
Oak 90 (6)
Chi 92 (4)
The Angels have 2 games left against one NL team, Wash. They have a 107 ERA+ and some good starters: Scherzer, Stasburg (if he is not on the DL) and Gonzalez.
Here are the players who have done it and the number of times. I used the Baseball Reference Play Index and set the minimum PAs as what qualifies for the batting title. If I lower that to 400 PAs (Trout might not make 502 PAs since he was hurt), McGwire and Williams would each get one more.