How to do continuous signal processing (i.e without windowing)? I'm working in speech recognition research, and I wonder if there is a way to analyse sound just like how ears do (i.e. without windowing), so for example to take some feature continuously with an update for each new sample.

It is similar to talk about windowing with overlapping ratio of 99% or something like this,  you can look at the question like: How does ear working without windowing concept?
Yes, it's possible to analyse sound the way ears do.

For example, you could compute the DFT of a signal continuously using several Goertzel filters.

$$
y_k[n] = e^{j2\pi k/N} y_k[n-1] + x[n]
$$

where $k= 0,1,\ldots, N-1$, so that $y_0$ is the DC or zero frequency term.

Of course, this is an unstable filter, so some resetting or forgetting factor is needed to keep it stable. There are ways to make it better behaved numerically (see pp84-93 of Stoer &amp; Bulirsch).

The reason why we don't generally do things this way is that we are usually trying to do them in the most computationally efficient way possible.  And nature doesn't necessarily do this, so we find ways to do similar things but that work better on a computer.