What is the highest angle the moon ever makes above the horizon at the North Pole? The question is in the title. I'd like to find out the maximum angle the moon ever makes above the horizon at the North Pole. By "the horizon at the North Pole," I mean the tangent plane to the earth at that point.

Note that I'm not asking for the highest angle that the moon ever appears to make above the horizon. Appearances are distorted by refraction, etc., although I'm not sure by how many degrees. Though I'd be interested in the exact distortion: is there a data repository where I can look up the highest observed angle? 

I would make the following prediction. The moon's orbital plane is inclined to the earth's equatorial plane at about $23.5+5\approx 28.5$ degrees. Let $R_e$ denote the radius of the earth and $R$ denote the radius of the moon's orbit around the moon. A little geometry for the highest point suggests

$$\theta_{\text{max}}=\tan^{-1}\left(\frac{R\sin(28.5^{\circ})-R_e}{R\cos(28.5^{\circ})}\right)$$

But because $R&gt;&gt;R_e$ (about 385,000 compared to 6300 km), $\theta_{\text{max}}$ should be just a little less than $28.5^{\circ}$.

Is this right?
${R_e}/{R}$ is about 0.017 radian or 1&deg;.
Using your trigonometry and these values:


Lunar orbit inclination = 5.15&deg; (varies between 5.0&deg; and 5.3&deg;)
Lunar perigee = 362600 km, apogee = 405400 km
Earth polar radius = 6357 km
Ecliptic obliquity = 23.44&deg;


I get 27.7&deg; at perigee and 27.8&deg; at apogee.
At that altitude, atmospheric refraction is only about 0.03&deg;.
The highest apparent altitude I found with JPL HORIZONS was 27.91&deg; on 1987-09-15 around 17:15 UT.

Due to nodal precession,
the Moon doesn't reach &plusmn;28.6&deg; geocentric declination every year, only near a major lunar standstill every 18 or 19 years, e.g. in 2006 or 2025.
Near a minor lunar standstill, e.g. in 2015, the Moon's declination is limited to &plusmn;18.3&deg;.