what factor limits the number of eclipses each year
The alignment of the Sun, Moon and Earth.
Introduction
This occurs when the Sun, Moon and Earth are aligned. 
UNANSWERABLE UNANSWERABLE Special conditions must occur for the two events to coincide because the Moon's orbit crosses the ecliptic at its orbital nodes twice every draconic month (27.212220 days) while a new moon occurs one every synodic month (29.530587981 days).
Solar eclipse
Introduction
Special conditions must occur for the two events to coincide because the Moon's orbit crosses the ecliptic at its orbital nodes twice every draconic month (27.212220 days) while a new moon occurs one every synodic month (29.530587981 days). Solar (and lunar) eclipses therefore happen only during eclipse seasons resulting in at least two, and up to five, solar eclipses each year; no more than two of which can be total eclipses.
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Solar eclipse [SEP] Introduction
A solar eclipse occurs when a portion of the Earth is engulfed in a shadow cast by the Moon which fully or partially blocks sunlight. This occurs when the Sun, Moon and Earth are aligned. Such alignment coincides with a new moon (syzygy) indicating the Moon is closest to the ecliptic plane. In a total eclipse, the disk of the Sun is fully obscured by the Moon. In partial and annular eclipses, only part of the Sun is obscured. If the Moon were in a perfectly circular orbit, a little closer to the Earth, and in the same orbital plane, there would be total solar eclipses every new moon.