Hello, forum peeps,
First Post here so I hope I can help.
Since there are very few concrete data points on the internet regarding credit line increases for this card I figured I would post my experience.
Aviator Red Card approved 9-2-2017 with a starting limit of $9,600. I did not call recon to increase it. FICO was around 702 and I had no previous Barclay cards or accounts.
I only got the card to churn as I got the 60k Bonus Miles for any one purchase. However, they immediately sent me a balance transfer of 15 months @ 0% with a 3% Fee. I couldn't resist so I transferred $7500 to the card and paid about $100 over the minimum payment due each month. I only put about $200 spend on the card the first week and haven't used it since.
I upgraded to the Silver Aviator in January 2018.
Today, on March 12, 2018, my account was considered to be 5 months old as my first statement didn't cut until Oct 2017. (Most people say you must wait 6 months for any APR reduction or soft pull CLI)
I called in to "ask about an APR reduction," in hopes that I would get a soft pull CLI. The lady who answered the call informed me that she could reduce my APR by 1%, down and give me a soft pull CLI of $4k. I told her I was looking for $25-30k total credit limit and authorized a HP. She transferred me to a credit analyst who looked at my credit and, after asking me some questions about my other CC balances, ultimately decided to give me an increase to $25k, stating the only reason he couldn't do higher is due to my account only being 5 months old. My highest card limit is AMEX Delta Reserve, which is $40k, and 6 years old.
FICO today is 762 with utilization of 20% (all of those CC balances are at 0-2.99% APR balance transfers)
I will put 1/2 of my income taxes on this card this year and all of my tax bill on the card next year due to the $3-6K EQD bonus get with $25-$50k annual spend, as I switched my flying from DL to AA this year.
All in all, I am pretty satisfied with the result of this.
And I hope this helps someone searching the internet for this info! 
This CLI finally puts me above the ever-elusive $200k Club in available personal credit!!! 
(I have another $40k in Business Cards between Chase and AMEX)
