Say you wanted to run a X point FFT on the last X audio samples that were played. The problem being, using a normal hann window function would place emphasis on the "middle" of the audio sample. However, the most current audio samples played (the audio samples toward the end) are of the greater importance for algorithms such as real-time beat detection. The question is, which window function best emphasizes the frequencies at the end of the waveform?
A full-width rectangular window (a.k.a no window) would best emphasize spectral content near the edges of an FFT aperture, at the cost of artifacts caused by convolution of the spectum with a Sinc function, also called "spectral leakage". If you can ignore or compensate for these spectral artifacts then there's little reason to window.
But you might be asking the wrong question. If you want the fastest possible response to some specific frequency band, you should be using a minimum phase filter (or a bank of them, as needed) or a close approximation to such, not a faster block DFT. Then you can update with each new sample instead of waiting to process a block, and tune the time-frequency response of the filter(s) for each specific band of interest.
I think the glitch in your question comes out at the very end: there are no "frequencies at the end" of the windowed signal. When performing the Fourier transform, the frequency bins don't have localized time; they describe the entire signal. But you knew that, that's why you are doing the STFT.
If you are doing beat detection, you could work in the time domain to get the absolute lowest latency. But if you are doing beat detection that also detects some transient in the frequency domain, where a sine tone shifting from 440 to 880 hz counts as an onset -- I'd suggest there's more of a fundamental limit to your latency, dependent on the pitch period; and in that case your best bet is to reduce the window size to, say, capture 4 periods of the lowest frequency you are interested in. If you are "weighting" the last bunch of samples coming in, you might as well shrink the window and get a predictable frequency response out of it. Another thought is to run parallel FFTs, one with a short window, and one with a long window...