No, you can't "up the voltage" but you can connect more than one LED per pin. Read about Charlieplexing here and elsewhere. Make sure you use current limiting resistors.
You can do it with a [u]shift register[/u] but it usually requires at least two pins. One for "clock" to tell it when to shift and one for "data" or to set-up the initial condition, and/or to clear.
But, it's possible to make a traditional chase effect by setting-up the initial condition, then make a loop by feeding the output (last register) back-into the input (first register). Or, you can make a "Johnson Counter" by inverting the output before feeding it back to the input. None of that takes a microcontroller, all you need is a shift register (or flip-flops) and a clock (like a 555 chip).
With 3-pins (data, clock, latch) you can address an almost-unlimited number of LEDs. I've used six 8-port [u]MAX6968[/u] LED driver chips to address/control 48 LEDs. This could have been done with 3-pins but it's a stereo sound-activated lighting effect with 3 daisy-chained driver chips on each side. The clock & latch pins are shared, but there are separate data pins for left & right so I used 4 output pins.
Or, with Neopixels you can address an almost unlimited number of LEDs with just one control line/pin.
If you think about it... The whole internet runs serially so "philosophically" all you need is one transmit pin and one receive pin. Or, all of cable TV comes over one wire and a cable-modem works over one wire...