As an engineering major with a high-credit-hours minor and another interest that requires a specific time slot a few times a week, my son has had a difficult time registering for more than one or two honors courses per semester on average.
It was easy for him to load up on honors courses during freshman year. Many of the courses that students take in their freshman year come with an Honors option. But after freshman year, my son found that most engineering courses did not come in an honors version, nor did most of the courses that were required for his minor.
And so, between the limited number of honors courses offered and the limited time slots left in his schedule after registering for his required courses, he found that he could often only register for 1 or 2 “straight-up” honors courses each semester for much of his degree. No matter – even with those restrictions, one can easily complete the honors requirements. But as to your question – is there any limit? – you may find that you’re limited by what is offered and when it’s offered.
Of course, there’s always the option to contract for honors credit (Honors Course Contracts) or do research for honors credit (Honors Independent Study). But as for “straight-up” honors courses that are offered on the TAMU Class Schedule, you may find that you are quite limited as you progress in your degree.