RF24 Only Works Once

if the power goes through the button, the resistor will limit the current available

one way to check if this is the cause would be to add a button to your arduino, keep the system powered and use the button to trigger the message (and wait for button release)

something like this (typed here, untested of course)

#include "Arduino.h"
#include <SPI.h>
#include <RF24.h>

// This is just the way the RF24 library works:
// Hardware configuration: Set up nRF24L01 radio on SPI bus (pins 10, 11, 12, 13) plus pins 7 & 8
RF24 radio(7, 8);

byte addresses[][6] = {"1Node", "2Node"};
const byte greenLed = 3;
const byte redLed = 4;

const byte buttonPin = 2;

// -----------------------------------------------------------------------------
// SETUP   SETUP   SETUP   SETUP   SETUP   SETUP   SETUP   SETUP   SETUP
// -----------------------------------------------------------------------------
void setup() {
  pinMode(greenLed, OUTPUT);
  pinMode(redLed, OUTPUT);
  pinMode(buttonPin, INPUT_PULLUP);

  Serial.begin(9600);
  Serial.println("THIS IS THE TRANSMITTER CODE - YOU NEED THE OTHER ARDIUNO TO SEND BACK A RESPONSE");

  // Initiate the radio object
  radio.begin();

  // Set the transmit power to lowest available to prevent power supply related issues
  radio.setPALevel(RF24_PA_MIN);

  // Set the speed of the transmission to the quickest available
  radio.setDataRate(RF24_2MBPS);

  // Use a channel unlikely to be used by Wifi, Microwave ovens etc
  radio.setChannel(124);

  // Open a writing and reading pipe on each radio, with opposite addresses
  radio.openWritingPipe(addresses[1]);
  radio.openReadingPipe(1, addresses[0]);
  // Ensure we have stopped listening (even if we're not) or we won't be able to transmit
  radio.stopListening();
}

void loop() {
  if (digitalRead(buttonPin) == LOW) { // button pressed
    delay(15); // poor's man anti-bouce
    unsigned char data = 1;
    Serial.println("About to transmit");
    // Did we manage to SUCCESSFULLY transmit that (by getting an acknowledgement back from the other Arduino)?
    // Even we didn't we'll continue with the sketch, you never know, the radio fairies may help us
    if (!radio.write( &data, sizeof(unsigned char) )) {
      Serial.println("No acknowledgement of transmission - receiving radio device connected?");
      digitalWrite(redLed, HIGH);
      delay(1000);
      digitalWrite(redLed, LOW);
    } else {
      Serial.println("Acknowledgement received");
      digitalWrite(greenLed, HIGH);
      delay(1000);
      digitalWrite(greenLed, LOW);
    }

    while (digitalRead(buttonPin) == LOW); // active wait until button is released
    delay(15); // poor's man anti-bouce
  }
}

(button would be on pin 2 set as input pull-up , wired this way: pin 2 <---> button <----> GND)