I can't seem to get a 155 degree mash (16 lbs. grain, 4.25 gals. water) to 170 degrees without adding a huge amount of water to mash out. I added 2.5 gals. at 195 degrees and only got to 166 degrees. I'm pushing the limits of my 11 gallon mash/lauter tun. What am I doing wrong?

The equations popularized by John Palmer for boiling infusions are useful here:

Vadded = ( ( Tdesired - Tcurrent ) * ( ( 0.05 * R ) + Vtotal ) ) / ( 212 - Tdesired)

Obviously the 212 in the denominator is boiling water. The 0.2 from the original equation changes to 0.05 so that we can use gallons instead of quarts.

There are a few assumptions you have to make here:

1.) Your current temperature is accurate, i.e. most people just use the temperature they mashed at in the calcs when it may have dropped lower over the course of the mash;

2.) That you are still at 212 degrees when you add the water.

So for your case:

Vadded = ( ( 170 - 155 ) * ( ( 0.05 * ( 4.25 / 16 ) + 4.25 ) ) / ( 212 - 170)

Vadded = ~ 1.52 gallons of boiling water

Let's try to apply a real world fudge factor here and assume that you lost 2 degrees over the course of the mash and that the boiling water cools a bit to 208 before infusing:

Vadded = ( ( 170 - 153 ) * ( ( 0.05 * ( 4.25 / 16 ) + 4.25 ) ) / ( 208 - 170)

Vadded = ~ 1.90 gallons of boiling water

A mashout is beneficial IF you get it into the 172 range and HOLD it for at least 10 minutes. Without bogging down anyone with the science behind it, the mashout/mashoff is a foam-enhancing mash step and holding it for a duration of 10+ minutes bolsters the production of compounds known as glycoprotiens, which are foam positive.