There are ways… but it would take some doing. I’m no EE, but I have played with small boost circuits for my illuminated tailcaps that can run off an AA or 14500.
I’ve used the TPS61099x (such as the TPS610993) to boost the voltage to a given output (eg: 3.0V) regardless of the input voltage. Runs fine down to 0.7V and will switch to pass-through mode if the input voltage exceeds the desired output voltage, so it’ll handle AA and li-ions just fine. Then use that boosted voltage as the input to any of our ATtiny’s. This chip is tiny, and works with just a few external components (also tiny). Very efficient and very low IQ.
Another option is one that I’ve never played with before, and you don’t see much talk about… the ATtiny43U. It has a built-in voltage boost circuit, though you’d still need the external inductor and capacitors (MC webpage, design guidelines).
This is all just providing a small amount of boosted juice for the ATtiny itself. This wouldn’t be for boosting the LED juice… just controlling the circuit that does so.