R, 71 bytes

f=\(n,m=2,l=0)`if`(n%%m,-sort(-c(if(l)l,if(n>1)f(n,m+1))),f(n/m,m,l+1)) 

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Recursive function that directly calculates prime exponents, and sorts them on every recursive call (which is unneccessary except for the outermost call, so overflows the stack for large inputs).

R, 74 bytes

\(n,f=\(n,m)if(n>1)`if`(n%%m,f(n,m+1),c(m,f(n/m,m))))-sort(-table(f(n,2))) 

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Uses recursive helper function f to calculate prime factors, and then sorts their exponents a single time (and thus copes with much larger inputs, but at the expense of 3 more bytes of code).

R, 71 bytes

f=\(n,m=2,l=0)`if`(n%%m,-sort(-c(if(l)l,if(n>1)f(n,m+1))),f(n/m,m,l+1)) 

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Recursive function that directly calculates prime exponents, and sorts them on every recursive call (which is unneccessary except for the outermost call, so overflows the stack for large inputs).

R, 74 67 bytes

Edit: -7 bytes thanks to pajonk

\(n,`?`=\(n,m)if(n>1)`if`(n%%m,n?m+1,c(m,n/m?m)))-sort(-table(n?2)) 

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Uses recursive helper function ? to calculate prime factors, and then sorts their exponents a single time (and thus copes with much larger inputs, but at the expense of 3 more bytes of code).