1) What voltage does your chip require?
Looking at the datasheet page 4, the analog parts of ESP8266 require 2.5-3.6V and the digital/IO parts require 1.8-3.6V. Thus it will work with a minimum voltage of 2.5V or 3V.
So you can use a 3.0V LDO or even a 3.3V LDO which will run in dropout mode (without regulation) when battery voltage drops below what is required to achieve 3.3V output. It is best to select a PMOS based LDO instead of a PNP one as the latter will have increased ground current in dropout when the pass transistor saturates. A PMOS based LDO in dropout mode will behave like a resistor, the RdsON of the internal FET.
Another feature you may find useful is a Power Good output to turn off ESP8266 when battery drops below 3V to protect it from overdischarge, but you may also do this in software if you use the ADC.
2) Power dissipation
With 4.2V in, 3.3V out, and 200mA current it will dissipate 180mW so a package capable of high dissipation is not necessary. A SOT-23 or SOT-89 package will be fine, small, and easy to solder.
3) Dropout voltage and regulation
Since ESP8266 will still work at 2.5V, for a 3.0V LDO you should aim for 2-300mV or lower dropout voltage at 200mA, leaving some margin for transient output voltage drop when the chip draws pulsed current. A fast transient response LDO will allow a smaller output cap.
Other criteria: fixed regulator since you say you want minimum parts count, low dropout, low idle current for battery life, stable with ceramic cap for low parts count and small size.
I've entered all this stuff in the mouser search and we have some candidates. Feel free to look at the list.
For example TLV70030 fits most criteria and has a fast transient response, you can check on the graph that the output voltage stays in the allowed range when output current is pulsed.
Some ultra low quiescent current LDOs look like they'd work but if you check the transient response, you will notice they are too slow so output voltage will drop below 2.5V unless you use a huge output cap. For example this is MCP1700, with very low 1.6µA quiescent current:
Notice the output voltage is displayed at 500mV/div, so it drops by 500mV on a 100mA output current pulse. Also the scope is set to 200µs/div. Compare these values to the TLV70030 above, for this one the scope is set to only 10µs/div and 50mv/div. So the curves look somewhat identical, but the result is completely different: one LDO is at least 10x faster than the other (at the cost of higher quiescent current), which means the output voltage stays in range with a tiny output cap. When the load draws current pulses, always check transient response with a scope. If supply voltage gets out of allowed range, your microcontroller will misbehave, crash or reboot.
A LDO with input voltage vs quiescent current curve like this should be avoided as it will draw extra current when the battery is discharged:
NCV8114 doesn't have this problem, it should be a good choice too.
So as you see it isn't all about dropout, it is about keeping the output voltage in the allowed range considering input voltage and output current variations. Please double check the rest of the parameters before choosing a chip, of course.