Is it possible to transmit using an NRF24L01+ without an Arduino?

The nRF24L01+ has a somewhat complex SPI based interface, where many registers have to be configured and tested and timeouts honored. Therefore it requires some kind of microcontroller to send or receive even the simplest message.

This can be as simple as a 3.3v Arduino Pro Mini (less than $3 with shipping), and there are libraries which simplify the task of interfacing to it - in particular see the RF24 library (there are many variations, see https://github.com/TMRh20/RF24 for a good one).

But many other microcontrollers could work as well.

Once you bite the bullet and accept the need for a microcontroller, yes the code can identify the source of the signal (which pushbutton) and much, much more.

Or you could use a dedicated device like an xBee which can be configured using a Windows utility to send the state of an input pin to another xBee; basically there is already a programmed microcontroller on the module along with the radio (not nRF24L01+ compatible).

An ATTiny85 can drive the NRF24L01+: "nrf24l01+ control with 3 ATtiny85 pins"

At least this is not a "full" Arduino...

Edit 2015-06-06:

Hackaday News: "EMBED WITH ELLIOT: MULTIPLEXING SPI USES FEW PINS"

The referred blog entry is: "nRF24l01 control with 2 MCU pins using time-division duplexed SPI".

(-: Nice countdown... 3 pins, 2 pins, ... ;-)

Note, there is also the NRF24LE1 SoC which has its own ULP MCU - good for transmitting simple data such as sensor readings. If used to Arduino, ARduino Pro Mini 3.3v and the NRF24L01+ is easier and better way to go. (Mod the Arduino to be ULP)