If you're looking for a more efficient way to randomize your array, I would recommend using a shuffle function instead of constantly selecting random indexes.

Keep in mind that this method assumes that the maximumAnswers value will always be less than or equal to the length of answerList plus one.

var answerList = ["answer 1", "answer 2", "answer 3", "answer 4", "answer 5"];
var correctAnswer = "CORRECT!";
var selectedAnswers = [];
var maximumAnswers = 4;

function generateAnswers() {
    var tempAnswerList = shuffle(answerList.slice()); // Make a copy of the answerList and shuffle it
    tempAnswerList = tempAnswerList.slice(0, maximumAnswers - 1); // Subtract 1 because we will add the correct answer
    tempAnswerList.push(correctAnswer); // Add the correct answer
    tempAnswerList = shuffle(tempAnswerList); // Shuffle again to randomize the correct answer's position
    console.log(tempAnswerList);
}

// Shuffle function source: http://stackoverflow.com/questions/6274339/how-can-i-shuffle-an-array-in-javascript
function shuffle(o) { 
    for (var j, x, i = o.length; i; j = Math.floor(Math.random() * i), x = o[--i], o[i] = o[j], o[j] = x);
    return o;
};

generateAnswers();

JSFIDDLE DEMO

I made some updates to your code because I am unsure about the necessity of using a large recursive function like the one you had:

var answerList = ["answer 1","answer 2", "answer 3", "answer 4", "answer 5"];
var correctAnswer = "CORRECT!";
var selectedAnswers = [correctAnswer];

function incorrectAnswers(answerList, number = 3) {
    //number represents the desired number of answers to be returned, with a default set to 3
    var wrongAnswers = [];
    //While the wrongAnswers array is not yet filled properly:
    while (wrongAnswers.length < number && answerList.length > number) {
        //Generate a random value
        var random = Math.floor(Math.random() * answerList.length);
        //Add it to the array if it is not already included (using indexOf)
        if (wrongAnswers.indexOf(answerList[random]) == -1) 
             wrongAnswers.push(answerList[random]);
    }
    //If the list of potential answers is shorter than the desired number, return all possible answers; otherwise return the generated list
    if (answerList.length <= number) 
        return answerList; 
    else 
        return wrongAnswers;
}

console.log(incorrectAnswers(answerList));

Check it out on Fiddle: http://jsfiddle.net/oybojgzm/2/

/*
    This function selects a specified number of random answers from a pool of possible answers and pushes them into a selected answers array.
*/
function selectRandomFromAndPushInto(count, answerPool, selectedAnswers) {
    //Create a copy of the answerPool to avoid modifying the original array
    answerPool = answerPool.slice();

    //Iterate 'count' times to select random answers
    for(var iteration = 0; iteration < count; iteration ++) {
        //Generate a random index within the range of the answerPool array
        var index = Math.round(Math.random() * (answerPool.length - 1));
        //Push the selected answer into the selectedAnswers array
        selectedAnswers.push(answerPool[index]);
        //Remove the selected answer from the answerPool to prevent duplicates
        answerPool.splice(index, 1);
    }

    //Return the selected answers array
    return selectedAnswers;
}

var answerPool = ["answer 1", "answer 2", "answer 3", "answer 4", "answer 5"];
var selectedAnswers = ["CORRECT!"];
console.log(selectRandomFromAndPushInto(3, answerPool, selectedAnswers));

Output may resemble the following when tested with node.js:

[ 'CORRECT!', 'answer 3', 'answer 5', 'answer 4' ]

To put it simply:

const choices = ["choice A", "choice B", "choice C", "choice D", "choice E"];
const correctChoice = "RIGHT!";
const chosenChoices = [correctChoice];
let randomIndex = 0;

function randomChoice() {
while(chosenChoices.length < 4){
    randomIndex = Math.floor(Math.random() * choices.length);
    if(!(chosenChoices.indexOf(choices[randomIndex])>-1))
        chosenChoices.push(choices[randomIndex]);
    }
}

randomChoice();

http://jsfiddle.net/oybojgzm/5/

One effective strategy is to initialize an array for testing purposes:

 var optionList = ["option A", "option B", "option C", "option D", "option E"];
 var correctOption = "CORRECT!";
 var chosenOptions = [];
 var randomNum, temp, temp2, length, index;

 function generateRandomOption() {
   temp=[];
   temp2=0; //number of elements currently in the temp array
   length=optionList.length; //store length value for efficiency
   while(temp2<3) { //continue loop until 3 numbers are in the temp array
     randomNum = Math.floor(Math.random() * length); //random index within range
     for(index=0; index<temp2; index++) //initial random number is always valid
       if(temp[index]==randomNum) //if index already exists in temp array
         break; //stop loop to avoid index matching temp2
     if(index==temp2) //true after the loop ends without breaking
       temp[temp2++]=randomNum; //store different indices in temp array
   }//end of while loop, obtaining 3 different random numbers
   //randomly insert the correct answer into the final array
   randomNum = Math.floor(Math.random() * 3); //index values from 0 to 2
   for(index=0; index<3; index++) {  //3 indices in the temp array
     if(index==randomNum) //if the position matches the random index for the answer
       chosenOptions.push(correctOption);  //place the correct answer in the output
     chosenOptions.push(optionList[temp[index]]); //add another answer to the output
   }
   console.log(chosenOptions);
 }