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Theoretically, there is no maximum or equilibrium constant that exists. However, there is a point where K is so large (the concentration of products is so much larger than the concentration of products) that the reaction is essentially running to completion. There is also a point where K becomes so close to 0 (concentration of reactants is so drastically larger than the concentration of products) that the reaction barely occurs and it is treated as such. Basically, there is no tangible limit to K, no calculations would be affected, and it's unlikely that we'll have to deal with extremely large equilibrium constants in this course (personally I have not seen anything beyond 10^-11 on the low side or 10^13 on the high side).
When K approaches very large or small values, essentially what is happening is that the reaction to going to completion or isn't happening at all. Therefore the reaction is not at equilibrium and you would not have an equilibrium constant. An example that Dr. Lavelle gave is class is strong acids. Strong acids completely dissociate, so they don't have an equilibrium constant since theoretically it is an infinitesimally large number.
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