New Order #1: How does this feel?

Haskell, 66 65 64 bytes

f n|n<3=n|m<-n-1=[k|k<-[1..],k`gcd`f m>1,all(/=k)$f<$>[1..m]]!!0

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Haskell, 60 bytes

((1:2#[3..])!!)
n#l|x:_<-[y|y<-l,gcd y n>1]=n:x#filter(/=x)l

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Zero-indexed; could save four bytes if series started with 2 (kinda (-1)-indexed, but without value for -1 being defined). Builds the infinite list by lazily maintaining the list of unused numbers.

Python 2, 104 bytes

This uses 0-based indexing.

from fractions import*
l=[1,2]
exec'i=3\nwhile gcd(i,l[-1])<2or i in l:i+=1\nl+=i,;'*input()
print l[-2]

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