look into Attiny45 or 85
Tinyy85 has 5 pins available after Vcc, gnd and reset. So that meets the 1 analog in an 3 digital out requirement. They can run on 3.3V and the 8 MHz internal oscillator. To program them you need to install a "core". I use the ATTinyCore for my tiny 85 projects. They are easy to bootload (to set fuses) and program using the Arduino as ISP sketch and an Uno (or other Arduino board) as a programmer.
I see there used to be an Arduino Gemma based on the ATTiny85.
If you multiplex the LEDs you could go with 11 pins for the 3 digits (8 for segments [7 seg, 1 dp] and 3 for digit select) or use 3 595 shift registers (one for each digit) would use 3 pins (clk, data out, latch).
Not sure the 595s would work. They test only up to 8mA and a total continuous current of 35 mA. I expect to be targeting 10-20mA per segment. The CD4511 seems promising, if my displays are common cathode (I hope they are )