How to do repeated requests until one succeeds without blocking in node?

There is no need to re-invent the wheel… you can use a popular async utility library, ‘retry’ method in this case.

// try calling apiMethod 3 times
async.retry(3, apiMethod, function(err, result) {
    // do something with the result
});

// try calling apiMethod 3 times, waiting 200 ms between each retry
async.retry({times: 3, interval: 200}, apiMethod, function(err, result) {
    // do something with the result
});

async GitHub page

async.retry docs