Question: What is the smallest unit of time?

Unit of time



A unit of time or time unit is any particular time interval, used as a standard way of measuring or expressing duration. The base unit of time in the International System of Units (SI), and by extension most of the Western world, is the second, defined as about 9 billion oscillations of the caesium atom.  The exact modern definition, from the National Institute of Standards and Technology is:
The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.[1]
Historically units of time were defined by the movements of astronomical objects.
Sun based: the year was the time for the earth to rotate around the sun. Year-based units include the olympiad (four years), the lustrum (five years), the indiction (15 years), the decade, the century, and the millennium.
Moon based: the month was based on the moon's orbital period around the earth.
Earth based: the time it took for the earth to rotate on its own axis, as observed on a sundial. Units originally derived from this base include the week at seven days, and the fortnight at 14 days. Subdivisions of the day include the hour (1/24th of a day) which was further subdivided into minutes and finally seconds.  The second became the international standard unit (SI units) for science.
Celestial sphere based: as in sidereal time, where the apparent movement of the stars and constellations across the sky is used to calculate the length of a year.
These units do not have a consistent relationship with each other and require intercalation. For example, the year cannot be divided into 12 28-day months since 12 times 28 is 336, well short of 365. The lunar month (as defined by the moon's rotation) is not 28 days but 28.3 days. The year, defined in the Gregorian calendar as 365.24 days has to be adjusted with leap days and leap seconds. Consequently, these units are now all defined as multiples of seconds.
Units of time based on orders of magnitude of the second include the nanosecond and the millisecond.
Historical

The natural units for timekeeping used by most historical societies are the day, the solar year and the lunation. Such calendars include the Sumerian, Egyptian, Chinese, Babylonian, ancient Athenian, Buddhist, Hindu, Islamic, Icelandic, Mayan, and French Republican calendars.
The modern calendar has its origins in the Roman calendar, which evolved into the Julian calendar, and then the Gregorian.

Scientific time units
The jiffy is the amount of time light takes to travel one fermi (about the size of a nucleon) in a vacuum.
Planck time is the time light takes to travel one Planck length. Theoretically, this is the smallest time measurement that will ever be possible. Smaller time units have no use in physics as we understand it today.
The TU (for Time Unit) is a unit of time defined as 1024µs for use in engineering.
The Svedberg is a time unit used for sedimentation rates (usually of proteins). It is defined as 10−13 seconds (100 fs).
The galactic year, based on the rotation of the galaxy, and usually measured in million years.[2]
The geological time scale relates stratigraphy to time. The deep time of Earth’s past is divided into units according to events which took place in each period. For example, the boundary between the Cretaceous period and the Paleogene period is defined by the Cretaceous–Paleogene extinction event. The largest unit is the supereon, composed of eons. Eons are divided into eras, which are in turn divided into periods, epochs and ages. It is not a true mathematical unit, as all ages, epochs, periods, eras or eons don't have the same length; instead, their length is determined by the geological and historical events that define them individually.
Note: The light-year is not a unit of time, but a unit of length of about 9.5 trillion kilometres (9,454,254,955,488 kilometres).
List

Units of time interrelated

All of the important units of time can be interrelated. The key units are the second, defined in terms of an atomic process; the day, an integral multiple of seconds; and the year, usually 365 days (more info HERE). Most of the other units used are multiples or divisions of these three. The graphic also shows the three heavenly bodies whose orbital parameters relate to the units of time.