Calculate A190810

Python 2, 88 83 72 bytes

You may want to read the programs in this answer in reverse order...

Slower and shorter still, thanks to Dennis:

L=1,;exec'L+=2*L[0]+1,3*L[0]-1;L=sorted(set(L))[1:];'*input()
print L[0]

Try it online

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This doesn't run as fast, but is shorter (83 bytes.) By sorting and removing duplicates each iteration, as well as removing the first element, I remove the need for an index into the list. The result is simply the first element after n iterations.

I may have out-golfed Dennis. :D

L=[1]
n=input()
while n:L+=[2*L[0]+1,3*L[0]-1];n-=1;L=sorted(set(L))[1:]
print L[0]

Try it online

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This version below (88 bytes) runs really fast, finding the 500000th element in about two seconds.

It's pretty simple. Compute elements of the list until there are three times more elements than n, since every element added may add at most 2 more unique elements. Then remove duplicates, sort, and print the nth element (zero-indexed.)

L=[1]
i=0
n=input()
while len(L)<3*n:L+=[2*L[i]+1,3*L[i]-1];i+=1
print sorted(set(L))[n]

Try it online

Jelly, 16 bytes

×3’;Ḥ‘$;
1Ç¡ṢQ³ị

Very inefficient. Try it online!

How it works

1Ç¡ṢQ³ị   Main link. Argument: n (integer)

1         Set the return value to 1.
 Ç¡       Execute the helper link n times.
   Ṣ      Sort the resulting array.
    Q     Unique; deduplicate the sorted array.
     ³ị   Retrieve its n-th element.


×3’;Ḥ‘$;  Helper link. Argument: A (array)

×3        Multiply all elements of A by 3.
  ’       Decrement the resulting products.
      $   Combine the two links to the left into a monadic chain.
    Ḥ     Unhalve; multiply all elements of A by 2.
     ‘    Increment the resulting products.
   ;      Concatenate 3A-1 and 2A+1.
       ;  Concatenate the result with A.

Python 2, 59 bytes

t={1}
exec'm=min(t);t=t-{m}|{2*m+1,3*m-1};'*input()
print m

Based on @mbomb007's Python answer. Test it on Ideone.