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#Python 2, 116 bytes

Meh. Not that short. The simple iterative solution.

L=[1,2,3]
n=input()
while len(L)<n:
 for i in range(2,n):
    if L.count(i)<L[i-1]and L[-1]!=i:L+=[i];break
print L[-1]

Try it online

Here's a really bad attempt at a recursive solution (128 bytes):

def f(n,L=[1,2,3]):
 if len(L)>=n:print L[-1];exit(0)
 for i in range(2,n):
    if L.count(i)<L[i-1]and L[-1]!=i:f(n,L+[i])
 f(n,L)