the moon - Is it possible to see a moonrise or moonset twice in a day?

This can only happen if the moonrise at a certain date is earlier than the moonrise at the previous day. There are two reasons why this could happen:

1. The body (Moon, Mars,...) moves in the opposite direction of the Sun viewed along the right ascension (or elliptical longitude) axis. The sidereal day is 4 minutes shorter than 24 hours which means that a static celestial body (a star) will rise 4 minutes earlier than the day before. However planetary bodies move (more or less) along the ecliptic, which is the Sun's path during the year along the celestial sphere. The Sun moves on average 4 minutes in elliptical longitude (and hence in right ascension with a bit more variation) every day. So for a body to rise earlier, the body would have to move slower along the celestial sphere than the Sun (or move in a retrograde direction). This is true for the outer planets (including mars) but not for the Moon, which moves in the same direction as the Sun and much faster (one orbit per month).


2. If the observer is at a sufficient high latitude on Earth (either north or south of the equator), the inclination of the celestial equator will be small enough so that a body that increases its declination rapidly enough will be able to rise earlier on the second day.

See for instance the figure below where each drawing is at the same time of day. The moon will increase its right ascension during the day (along the celestial equator downwards, the red arrow in the figure). If the observer is on the Earth's equator, moonrise will be later the next day. It does not matter whether the declination (the position above or below the celestial equator (to the left or right in the figure)) will increase or decrease. All objects with the same right ascension will rise at the same time on the equator. If the observer is not on the equator but near one of the poles (for instance at 65° latitude north), the increase in declination may be sufficient to counteract the increase in right ascension (see bottom panel). Note that even though the right ascension increases (the moon moves left along the celestial equator axis), the Moon will rise earlier during the day.

This only works (in the Northern hemisphere) when the Moon increases its declination (the height above the celestial equator) very rapidly. This happen only when the Moon is near the Vernal Equinox (i.e. in Aries or Pisces). And the observer needs to be at a high latitude (i.e. near the North pole such as Thule as mentioned by @barrycarter)