Samsung Galaxy S6
The Samsung Galaxy S6 is an ideal candidate as a rival as it's the same price as the phone, yet does a whole lot more. The camera and screen are better than that offer by HTC, and the size isn't that much larger either.
The power is off the chart in comparison, and battery life doesn't seem to have suffered as much as you'd expect given the thinner dimensions.
The interface is more garish and it's stuck on Android Lollipop - but there are upgrades coming that will probably make the battery even better. A great phone that it's hard to argue against unless you're really opposed to the overlay or sharper glass/metal design.
Moto X Style
In fairness, this is a phablet vs a 'normal-sized' phone, but in price there's a lot of similarity here. The power isn't that far off either - Snapdragon 808 vs 617 will offer relatively similar power output in a few areas, and both have a battery life that will last until around twilight each day.
The X Style is slightly cheaper though, and has a much better screen in terms of resolution - it's just whether you fancy the larger size. Both have a less-touched interface in terms of the manufacturer messing too much with it.
HTC has a brand new flagship phone on the market that was released about half a year after the HTC One A9. We really liked the HTC 10 while it's still not perfect, but if you're a fan of the HTC brand it's worth taking a look at whether this slightly more powerful phone is for you.
It comes with a thicker yet premium looking design, a 5.2-inch 2K display, 4GB of RAM, a Snapdragon 820 processor and a 3000mAh battery. It'll come with Android 6 Marshmallow software pre-loaded on your phone and you'll get the choice of either 32GB or 64GB storage options.
I think it comes down to the design of the phone though on whether you'd prefer this over the HTC One A9. If you're after that thinner, iPhone-esque design this won't excite you as much as the One A9.
I'm only putting this in the list as the design similarities have led to three different people asking if I was holding an iPhone when using the One A9 when out and about.
The iPhone has an inferior screen, a much higher price tag and a simpler (and less impressive) camera. The battery life is better, and many prefer the interface and appreciate the larger selection of high-quality apps.
However, there's not really a comparison here. If you're in the iPhone camp, the current crop is perfectly decent and probably enough to entice you to stay. If you are thinking of switching, it's only the lower price of the One A9 that will lure you here - otherwise, it's just the design similarity.
Sony Xperia Z5 Compact
In my eyes, the biggest competitor. It's got a reasonably similar-sized screen, the same price and is the little brother to a flagship.
In terms of power, battery life and camera though it's got the One A9 licked - it's a little chunkier in the design stakes and the interface is much uglier compared to naked Android (which the One A9 mostly has) but in terms of what you can do with the thing, the Z5 Compact seems to be a much more compelling buy.