Every class states your are eligible to receive the Ability Score Improvement (or a different feat) at level 4.

If I multiclass, and never go higher than level 3, does that mean I can never get an Ability Score Improvement unless I go to level 4 in a class?

e.g. FTR3/WIZ3/RGR3  is 9 levels, however, I am not given the opportunity to acquire the ASI.  Correct example?

Quote from Character77006 >>

Every class states your are eligible to receive the Ability Score Improvement (or a different feat) at level 4.

If I multiclass, and never go higher than level 3, does that mean I can never get an Ability Score Improvement unless I go to level 4 in a class?

e.g. FTR3/WIZ3/RGR3  is 9 levels, however, I am not given the opportunity to acquire the ASI.  Correct example?

 Correct (other than getting your level 1 feat if you are playing 2024 rules)

You're right. Ability Score Improvements are class features, gained when the class level grants them.

It's worth mentioning that Fighters and Rogues get more ASIs than other classes, not just at the usual levels.

EDIT: ninja'd by Jegpeg!