Three records that WILL be broken:

Most ABs in a single season without a sacrifice.


Most ABs in a career without a sacrifice.


Most career starts - as a pitcher - without a single complete game.



and a related record that WILL NOT be broken:

Most career starts - as a pitcher - without a relief appearance - currently held by Tom Glavine with 682.

Walter Johnson's career record of 110 shutouts. Whoops, I just realized that the OP is looking for records that CAN be broken. I suppose almost any record can be broken, but this WaJo record won't be.

Three records that won’t be broken IMHO:

DiMaggio’s 56 game hitting streak
Ryan’s career strikeout total
Ryan’s 7 no hitters. Most pitchers don’t even get one in their entire career. Can you imagine 7?!!

Career saves. If an ace reliever were to be identified at a fairly young age (25?) and pitched for a consistent top level team for many years racking up 40-50 saves per year, I could see Kimbrel challenging if he stays the same level for 8-10 more seasons.

Single-season WAR is in play thanks to Mike Trout who appears to be a multi-talented "WARmachine". Also, his new teammate Ohtani is challenging single season-hitting/pitching records (home runs and wins and pitching and offensive WAR combined marks).
Single-season Saves might be a possibility -- but managers are increasingly scared to wear down closers so it could be tough even for the best closers on the best teams.
Career marks that seem imperiled based on the current game include batters Strikeouts (although it would take a long career even from Aaron Judge)
Career strikeout rate hitters and (maybe) pitchers. Career saves is not inconceivable if someone can stay strong for a long time (which is very hard in the closer business) because of the high usage of closers
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ALSO Any records that have to do with MONEY -- career earnings, single-season salary etc.. Bryce Harper, Kershaw and Trout are all Free Agents soon their salaries may make ARODs salaries look small.

Relief pitcher stats could easily be broken. As mentioned earlier career saves, single season saves, most games pitched in a season/career.

Also fielding stats such as put outs, fielding % will always be broken as fields are immaculately groomed and glove technology is amazing.

Strikeouts. Finally, something I can excel at.

Is there a record for most pitches swung at without making contact? And if so, how do I get signed?
I'm pretty sure I could set that one easily.

Considering the record for WAR is 20.2 and Trout's best is 10.5...

Of course, that's unfair. The 20.2 was by Tim Keefe in 1883. If we go with hitters, the record is 14.1 in 1923* by Babe Ruth. That's still 3.6 better than Trout's best. I'd say that record is safe.

* - The fact that 1923 is Ruth's highest WAR total just shows you how dumb/wrong WAR can be at times. No one in their right mind thinks 1923 was his best year. Most would say 1921. Or 1927. Or 1920. But 1923? No. Yet WAR says it was his "best".

Interesting because I would think these, except perhaps DiMaggio's, are more likely to be broken now than a lot of other records. Records tend to be broken when the way the game is played changes. Strikeouts are much much more common now than they once were, both for pitchers and hitters, due to reassessments of their value/detriment. For batters, the current view is that you are better off swinging hard and with an uppercut even if you strike out more. Whereas for pitchers, the current thinking is that all pitchers are equal once the ball is put in play, and so the best measurement of a pitcher's value is his strikeouts. So if you can't throw mid-90s they don't even want to scout you. I'm not saying I necessarily agree with this thinking, and I do think it makes the game boring at times, but that is the current thinking. And that thinking lends itself to strikeouts and no-hitters. Scherzer, for instance, seems to flirt with a no-hitter ever time he's out there. The counter to this is that the current thinking is that pitchers shouldn't throw more than 120 pitches a game, or pitch more than every fifth day, which could make it harder to break those records. But there's still a chance. The records that will almost certainly never be broken are those for which changes in the game have completely put them out of reach. For instance, in the deadball era, parks were big, bumpy, and often in weird configurations, and players were playing with tiny gloves. That, and the fact that no one was hitting home runs, led to a lot of triples. Sam Crawford holds the record with 309 career. You are never going to see that broken in modern parks with the kind of fielding we have today. The current active leader is Jose Reyes with 128, and you can bet he's a lot faster than Crawford was. But the game is just different.