One way to do it is to get a Bachelor’s in one or two of the majors, and a masters in one or two others, or even a Ph.D. in one that wasn’t covered by the BS or MS. There is no requirement that your masters or Ph.D. be in the exact same field as your BS, especially when, as you point out, there is a lot of overlap in the course requirements. The advantage here is that you could then possibly finish your BS in 4 years, all on scholarship if you are so fortunate as to get one, and then for a Ph.D. the cost is often zero because you will be utilized as a TA in many schools. At least when I was getting my Chem Ph.D. they waived all tuition and fees and paid me a high enough stipend to afford basic food and housing. Hopefully that is still true. For a masters, it is much harder to get financed for free or reduced costs, but sometimes possible. Certainly beats taking 6 years to get a BS.
To be clear: Even if this were possible to do, I think it’s completely unnecessary and a bit silly, but since you are asking about implications I will address those.
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Just because you can theoretically slot in the number of required hours into a spreadsheet doesn’t mean those classes will fit together. Colleges schedule classes based on what they think students will take together/at the same time and what students won’t be taking. Since nobody expects engineering students to double-major - least of all with other engineering disciplines - you may find many of your required classes are scheduled at the same time, extending your time to degree by a lot. And not by just an additional year, but perhaps by 2 or 3.
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Most scholarships and loan programs only cover 4 or maybe 5 years of college, and/or are limited to a certain number of credits, which you will almost certainly exceed if you try to take on two additional engineering majors. Even one engineering major can bring you very close to those upper limits. This is a prime consideration if you are planning to finance college mostly through scholarships and loans.
Relatedly, you said you were planning on financing school partially through work. But if you are planning on three engineering majors, you will have very little time to work - you’ll be lucky if you can pull off 10 hours a week.
I think the big pending question here is why? For what purpose would you triple-major in three very similar fields, two of which are very very similar? You say that you really want to learn about things you are passionate about, but it is fully possible to do that without majoring. You seem to be under the impression that if you minor or just take some classes that you won’t get the “full range of knowledge,” but you’re mistaken - you can derive a very broad knowledge of an area from taking a few classes, especially if the area is one that is directly and very closely related to your major.
Besides that, a major locks you into a specific selection and sequence of classes. If you don’t triple major, you give yourself the freedom to take whatever classes you want to delve into the subjects you’re interested in learning about.