How to make responsive textbox expression start at a given frame?

Text animators can really foul up sourceRectAtTime() values unless you add a time value you want to use. If your text box is supposed to grow as the animation proceeds, then you have to define a variable to go in the parentheses that match the timing of the start or end Animator Range Selector so that the shape layer grows with the animation. To give you an expression that works, I need to see all of the modified properties in your text layer (press 'uu') and understand your workflow better. 

In this example, I'm just reporting the width of the text layer using a simple expression.

By tying the Animator 1/Range Selector/Start value to the width of a Shape Layer Rectangle, I could create a text box that followed the left edge of the first letter "T" and grew as the other characters appeared.

You would need to use an ease() expression and determine the values when said expression is supposed to start and end referencing the keyframes of your counter slider oir whatever else you are using. You may also need additional keyframing on the shape or otherwise it just pops in.

Mylenium