In the latest iOS version there's an iOS supplied form of auto-complete that works so well for me in many uses (but which does not auto-complete on a value list) that I stopped using the following work around in some of my solutions, but the following work around is something that I still us where appropriate:
You can set up a "search portal" where you enter text into a text field and a filtered portal of options listed below updates to list only entries that either start with or contain the search text. When you see the listed option that you want in the portal, you tap the portal row (where the field has been set up as a button to perform a script), to select that value. This can be placed in a popover button to make this a bit nicer set up on larger iOS screens such as the iPhone 6 (or larger) or an iPad. On smaller screens, you have to do a bit of "peek a boo" between the virtual keyboard and the portal (I first set these up when using an iPhone 4s and it's dinky screen...), but it does work.
For actual examples of this method, see:
"Adventures in FileMaking #2 - Enhanced Value Selection" This file is set up for use on a computer using FileMaker 13 so that you can open and see how things work "under the hood", but the filtered selection portals work pretty well in FM GO. There's also a "date picker" in a popover that provides the same function as the drop down calendar for entering dates on iOS devices that you can export from a container file if you are interested. It's been designed to be pretty simple to import into existing solutions without a lot of "fiddling" in order to get it to work.
Apologies if you are reading responses in your email. I forgot to add the code for the download link to enable downloading the example file. I've edited the original post to correct that link, but you'll have to return to the forum to get to and click it if interested.