As a metrologist, I am glad of this interest in correct notation, often not enough pondered also among metrologists but essential for understanding with each others.
I would say first that any numerical value of an experimental result is always expressed as a rational, not irrational, number, because the number of digits is always bounded by the position of the measurement uncertainty level, e.g. 1.2345(x) or 1.23456(xx), according to guidelines for correct notation of uncertainty –where x indicates the uncertainty applied to the least significant digit(s). However, it is true that the numerical value of the measured quantity is in itself an irrational number, unless we have a specific reason to understand that is a rational one.
In this respect, the numerical value assigned to any ‘fundamental constant’ (e.g. the Planck constant), whose numerical value we can only obtain through measurement, is always a rational number with an associated uncertainty (though sometimes the latter indication is dropped, in which case it is correct to add the three dots to indicate an irrational number).
In other words, all experimental numbers are ‘uncertain numbers’.
However, when used in a definition they must instead be written as exact numbers: for this purpose they are ‘stipulated’, indicating with that term the decision taken to agree on an exact value (it is a consensus value that does not modify at all the intrinsic uncertainty of our knowledge).
So it is 273.16 K exactly for the triple point of water. So is 299792458 m/s exactly for the speed of light in vacuum. So it would be for any other constant, should it be stipulated in future. Incidentally, for the speed of light the notation can be misleading: it is not an integer number, as it would become clear when writing it as 299792.458 km/s.
Also the value 273.15 K in the definition of the Celsius scale (t/°C = T/K – 273.15) is an exact one. This a tricky case. In fact, the definition implies that, when T = 273.16 K, thus t = 0.010 °C exactly. On the other hand, when one measure T = 273.1500(x) K, thus t = 0.0000(x) °C: however, in fact, this is not anymore necessarily the freezing point of water, exactly for the same reason why the normal boiling temperature of water is no more 100 °C but 99.974(x) °C.
Should in future temperature unit be defined using the Boltzmann constant, also the temperature of the triple point of water will not anymore be exact, not being anymore used in the definition of the kelvin. However, in order to avoid a discontinuity in the more precise temperature measurements due to the change between the old and the new unit, it should be assumed, at least initially, that the value will be 273.16000(15) K, where the uncertainty in parenthesis arises from the uncertainty of the Boltzmann constant determinations (1 ppm). Obviously, future determinations could depart from that value, since the value 273.16 K was stipulated before 1954, when the precision of the measurements was lower than 0.0001 K.