Not reaching target final gravity

I've been all grain brewing for about 20 batches now. I have often (with many different beers) run into the problem of not fermenting down to my target final gravity. For example I recently brewed a Hefeweisen using 5.50 lbs of 2row and 4.5 lbs of flaked wheat. The original gravity was 1.047 and the target final was 1.013. I pitched at 65F with a big starter. The fermentation was rolling very strong by the next morning and the temp was up to 68F. Fermentation had pretty much stopped by the fifth day and I checked the final gravity on day 15. I reached 1.023. Substantialy higher than my target of 1.013. I have tried in the past to restart batches like this with fresh yeast. In the end I got no more fermentation. I'm thinking that I must be producing too many unfermentable sugars some how. If this is the case what could I be doing to cause this? I am mashing in at 154F and mashing out at 168F.

Are you sure you are mashing at 154? A few degrees can make a big difference. Maybe verify with another thermometer, or use a calibrated thermometer to begin with.

+1 to mashing too high

Also, how long did you mash for? Even if you were at exactly 154F, too short a mash could leave you with a lot of unfermentables.

Bring some water to a boil and check the temp with your thermometer, is it 212°?

Now you know how far off your thermometer is. Assuming it is a dial type, calibrating it is as easy as turning the nut on the back of it so that it reads correctly.

Brew another batch after calibrating and see how it goes.

You may also want to account for altitude and ion content, depending on where you are and your water.

Use distilled water in your test, and with an average elevation in your state being 500 foot above sea level, at 500 foot above sea level your boiling point is 211°F, verify your elevation and Google it for the boiling point at your elevation. if your thermometer is reading below that number you have a good indicator that is your issue.