How do I set a minimum upper bound for an axis in Highcharts?

Highcharts doesn't seem to have an option for doing this at chart creation time. However, they do expose a couple methods to interrogate the extremes and change the extremes, getExtremes() and setExtremes(Number min, Number max, [Boolean redraw], [Mixed animation]) found in their documentation.

So, a possible solution (after chart creation):

if (chart.yAxis[0].getExtremes().dataMax < 10) {
   chart.yAxis[0].setExtremes(0, 10);
}

yAxis[0] references the first y-axis, and I'm assuming that you only have one axis in this case. The doc explains how to access other axes.

This isn't ideal, because the chart has to redraw which isn't too noticeable, but it's still there. Hopefully, Highcharts could get this sort of functionality built in to the options.

A way to do this only using options (no events or functions) is:

yAxis: {
    min: 0,
    minRange: 10,
    maxPadding: 0
}

Here minRange defines the minimum span of the axis. maxPadding defaults to 0.01 which would make the axis longer than 10, so we set it to zero instead.

This yields the same results as a setExtreme would give. See this JSFiddle demonstration.

Adding to Julian D's very good answer, the following avoids a potential re-positioning problem if your calculated max varies in number of digits to your desired upper bound.

In my case I had percentage data currently going into the 20's but I wanted a range of 0 to at least 100, with the possibility to go over 100 if required, so the following sets min & max in the code, then if dataMax turns out to be higher, reassigns max up to it's value. This means the graph positioning is always calculated with enough room for 3-digit values, rather than calculated for 2-digits then broken by squeezing "100" in, but allows up to "999" before there would next be a problem.

var chart = new Highcharts.Chart({
    chart: {
        renderTo: 'container',
        events: {
            load: function(event) {
                thisSetMax = this.yAxis[0].getExtremes().max;
                thisDataMax = this.yAxis[0].getExtremes().dataMax;
                if (thisDataMax > thisSetMax) {
                    this.yAxis[0].setExtremes(0, thisDataMax);
                    alert('Resizing Max from ' + thisSetMax + ' to ' + thisDataMax);
                }
            }
        }        
    },
    title: {
        text: 'My Graph'
    },
    xAxis: {
        categories: ['Jan 2013', 'Feb 2013', 'Mar 2013', 'Apr 2013', 'May 2013', 'Jun 2013']
    },
    yAxis: {
        min: 0,
        max: 100,
        title: {
            text: '% Due Tasks Done'
        }
    }
    //Etc...
});