I'd say our greatest strength is trouble-shooting, and system perfection/refinement. because our main drive is supposed to be (and appears to be) modeling reality as perfectly as possible.
Sometimes what we want to model perfectly is an idea or creative vision we have, which is basically a process of recreating something we already see in our heads, or building a construct around some basic principles we have a good feel for. I'd say this is the ultimate 'purpose', in that it'd be a result of good personal development and would provide a lot of satisfaction - externalising the internal, and being able to share it.
But overall, troubleshooting, really. It seems to be at the heart of every INTP's passion or at least daily life, aside from the creative stuff - error alert and then repair with least possible damage to the system. This lies behind all the advice we seem to find ourselves giving as well (modeling the entire situation, finding the weak/wrong points, and then suggesting a solution within the parameters [principles, moral concerns, etc] to fill the gap - usually failing to incorporate the complications present from being emotionally involved in the situation). Programming is, afaict, very creative and very precise troubleshooting. Teaching, another thing a lot of us seem to have ended up in, involves a lot of that as well (identifying the exact problem in the student, isolating it and then providing a solution for improvement). You might also call it 'variable juggling' - satisfying the parameters, trying to use as general a principle as possible while still being accurate and always searching for a better, even more encompassing structure that can hold all the constant change in place and solve for all possible values. This root seems to be in most of our experiences of satisfactory work/life, though the fields are very varied and the way it manifests may not be obvious.
Seems to be how Ti-Ne works, anyway. Constant weeding and refinement on a conceptual level. Obviously that's my model of it.
And I've tried to express it and show how it extends to actual life but of course it's always just a shadow of what I mean and see in my head.
*Actually*, I'd like to revise the 'ultimate purpose'. I think the ultimate purpose would be to perfectly express and communicate our models, whatever they are. Precisely and concisely. It's always so exciting to read writing which has channeled the same crystal you see more perfectly, and managed to capture all its angles at once. When I do anything like that it's always incredibly satisfactory (also relieving because it usually feels like a burden when not expressed properly). The verbal model is complete.