If you're just looking for interesting material, it's really hard to say what will interest you: obviously, even mathematicians have their own preferences. If you want to learn higher mathematics, you will almost definitely need to learn calculus, as it is an essential tool to many disciplines, even those you won't see while studying engineering. I would suggest a good background in single-variable and multi-variable calculus, ordinary differential equations, and linear algebra before attempting to study differential geometry, as it generalizes the techniques of calculus and applies them to more abstract constructions: beginning with surfaces and moving up to manifolds (and ODEs and linear algebra are simply very useful for understanding and problem solving). If you can, I would even suggest learning real analysis before studying differential geometry, although it isn't necessary, because it will make you more familiar with the proof techniques and theorems you'll see in DG. (But you can probably get by with minimal linear algebra and a decent understanding of multivariable calculus if you don't have time for the rest of my recommendations.)
Personally, I enjoy the algebra/number theory side of mathematics a bit more than the analytical side, so I have to recommend taking a number theory course or an abstract algebra course. Number theory is nice because a lot can be introduced with minimal background knowledge, but before long the problems start drawing from surprising areas of mathematics (although a first course most likely won't expose you to the applications of complex analysis in number theory). Abstract algebra is doable, but make sure you're comfortable with proof, as it is a large part of any algebra course (and any higher math course in general). Both these areas seem fitting, as a first exposure to either doesn't require too much background in other higher maths, even calculus. Be warned: these areas are probably going to have less applications to your engineering studies than some of the others, but if I'm not mistaken, this is the type of thing you're looking for.
Complex analysis is also an interesting area with many applications, both to the practical engineer and the pure mathematician. In a way, it's a bit more rigid than real analysis, because the requirements on complex functions are much more stringent: it takes more to make a respectable function here than it does in the purely real case. With complex analysis, one can solve a lot of problems that aren't able to be solved by purely real methods: for example, certain integrals that just weren't possible before become a breeze, and some algebraic problems become much more natural when you pass to a complex analogue. However, you might wind up needing to take a complex analysis course for your major anyway, so perhaps this isn't quite what you're looking for.
I would suggest waiting to look at algebraic geometry, as you need a strong background in a lot of other mathematical fields (especially abstract algebra) to really get a hold on it. I haven't taken a course yet, but I have spoken to others who have: and the consensus is that it is hard, and that if one tries to take it too early, the ideas might not sink in. If you take abstract algebra and decide that you find that interesting and want to do more, then perhaps look into a number theory course that uses abstract algebra and/or an algebraic geometry course.
As Jim said, topology is another option that doesn't have that many prerequisites, but you have to have a certain level of mathematical maturity (as you must in any upper level math course). In some sense, topology is like "rubber geometry:" it studies what properties stay the same under continuous transformations (although there's much more to it than that!). If you think you might be interested in topology, start with a course on point-set topology to get the basics, and then move on to differential or algebraic topology, depending on whether you like analysis or algebra more.
Best of luck in your mathematical pursuits!