I thought I might as well throw my hat in the ring, using ES6 Promises...
function until_success(executor){
    var before_retry = undefined;
    var outer_executor = function(succeed, reject){
        var rejection_handler = function(err){
            if(before_retry){
                try {
                    var pre_retry_result = before_retry(err);
                    if(pre_retry_result)
                        return succeed(pre_retry_result);
                } catch (pre_retry_error){
                    return reject(pre_retry_error);
                }
            }
            return new Promise(executor).then(succeed, rejection_handler);                
        }
        return new Promise(executor).then(succeed, rejection_handler);
    }
    var outer_promise = new Promise(outer_executor);
    outer_promise.before_retry = function(func){
        before_retry = func;
        return outer_promise;
    }
    return outer_promise;
}
The executor argument is the same as that passed to a Promise constructor,  but will be called repeatedly until it triggers the success callback. The before_retry function allows for custom error handling on the failed attempts. If it returns a truthy value  it will be considered a form of success and the "loop" will end, with that truthy as the result.  If no before_retry function is registered, or it returns a falsey value,  then the loop will run for another iteration.  The third option is that the before_retry function throws an error itself.  If this happens, then the "loop" will end, passing that error as an error. 
Here is an example:
var counter = 0;
function task(succ, reject){
    setTimeout(function(){
        if(++counter < 5)
            reject(counter + " is too small!!");
        else
            succ(counter + " is just right");
    }, 500); // simulated async task
}
until_success(task)
        .before_retry(function(err){
            console.log("failed attempt: " + err);
            // Option 0: return falsey value and move on to next attempt
            // return
            // Option 1: uncomment to get early success..
            //if(err === "3 is too small!!") 
            //    return "3 is sort of ok"; 
            // Option 2: uncomment to get complete failure..
            //if(err === "3 is too small!!") 
            //    throw "3rd time, very unlucky"; 
  }).then(function(val){
       console.log("finally, success: " + val);
  }).catch(function(err){
       console.log("it didn't end well: " + err);
  })
Output for option 0:
failed attempt: 1 is too small!!
failed attempt: 2 is too small!!
failed attempt: 3 is too small!!
failed attempt: 4 is too small!!
finally, success: 5 is just right
Output for option 1:
failed attempt: 1 is too small!!
failed attempt: 2 is too small!!
failed attempt: 3 is too small!!
finally, success: 3 is sort of ok
Output for option 2:
failed attempt: 1 is too small!!
failed attempt: 2 is too small!!
failed attempt: 3 is too small!!
it didn't end well: 3rd time, very unlucky