It is a bit hard to answer that as your command of the language does not seem to be so good.
On research, I gather you are working on logging domestic or business electricity usage.
Saying "microcontrollers that I can use with Arduino" makes no sense as in general, an Arduino such as your Mega2560 already is a microcontroller device and "connecting an ESP8266 to them as a stand-alone" makes no sense either as the ESP8266 is already a microcontroller in its own right with as I recall, substantially more processing power, RAM and Flash memory than the Mega2560 and is "dirt-cheap".
So to ask your question, you need to specify for this particular project, exactly what it is, what form of processing you require, how much data you wish to store and over what timeframe, how that data is to be acquired and how it is to be disposed to which other system?
I can't directly help you but any microcontroller can run "stand alone".
There are a couple of BIG advantages to the Arduino - It's a complete board that's used as the final main board, and the bootloader means the board is also the development platform.
A single board computer (such as the Raspberry Pi) shares these advantages, plus you get an operating system and (if you want/need them) you can use standard peripherals like a keyboard, mouse, and display, etc., without writing your own drivers or using custom-specialized software.
A single board computer is probably less convenient than a microcontroller if you're connecting custom or "simple" hardware like LED strips & switches.
If you use a "regular-random" microcontroller chip you'll generally need to buy a development board and a programmer and you may have to pay for the IDE. Then, you need to design your own board to go into the product/project.
The chip itself isn't usually too expensive unless you need something high-powered like they use in a high-end smart phone or the microprocessor used in a computer.