Brouwer's analysis of the continuum depends heavily upon the notion of a "free choice sequence" (aka a lawless sequence). The idea is that a creating subject -- i.e., you or me -- is free to make an unbounded sequence of choices to produce a sequence. At any finite time, you can only have made a finite number of arbitrary choices, but if you wait long enough you can make any number of arbitrary choices. Because these choices are arbitrary, free choice sequences cannot be described as being generated by any finite rule.
This is a constructive process, if you imagine (for example) generating a lawless sequence by throwing dice or from the clicks of a Geiger counter. But it's not a finitist one, because there is no finite data from which a choice sequence can be computed. This let Brouwer retain the idea that a mathematician can only have considered a finite amount of information at any point, without having to make any identification of the real numbers with computable functions.
In terms of complexity, there's an interesting analogy between choice sequences and the question of whether randomization gives you extra power -- the Miller-Rabin primality test lends immense force to the intuition that RP is stronger than P, but we might be centuries from proving it.
Even if we set aside randomness, though, an ultra-finitist/complexity-aware-logic is something that basically every logician I've ever talked to has agreed should exist, except we basically don't know how to do it. It's something that pretty clearly demands more creative insight than technical grunt, and unfortunately imaginative leaps don't come on demand. :/