Starting from when I had no credit reporting (late 2015). My ex wife signed me up for the Buckle card to buy clothes for my daughter. It's stayed at $1300 ever since then. I went for the Cap One Union Plus MC 8 months later and that's sat at $2500 ever since. Discover was next, starting at $5500 and was my first card to break $10K. 3 months later I app spreed and the BoA AAA Rewards card I picked up then blew all my other cards into the weeds. Next after BoA, Chase FU broke $10K via 2 SP CLI's and a little help from my CSP. Last but not least, I just recently picked up Savor with a cool $10K SL.
First card I manage to"grow" to $10K was Walmart card. Closed it a month later 
First approval over $10k was BECU cash back visa.
My first credit cards were when I was 18 or 19, and they had $200 or $500 limits. Several store cards, a few majors, and always maxed out. The highest card I had was probably $2,000 from Discover as I got into my 20's. I missed payments. Fees push me to owing even more than my credit limit is. I eventually canceled all of them when I was around 28 and joined a credit repayment program where I sent them money and they negotiated a lower interest rate and paid off the cards. It took 4 years to pay everything off (I think I had 10 cards for this round).
The next time that I got credit cards was right after paying off the old cards. Highest limit this time was $3700 from Menards. I was very impressed that they gave me that much! I ran it up fairly quickly and then got a Lowes card and ran that up too, then a Home Depot card.. and on it went. I got a few majors too, but no substantial limits. A few were $1,000 or $2,000 but I kept maxing out and applying for a new card until all I could get was $200 cards again and then denials. More missed payments. More fees. I hadn't learned my lesson. After just a couple years, I canceled everything and joined the program again to pay everything off. It'll take another 4 years to pay it off.
My air conditioner goes out while I'm still repaying. The repayment company had told me that I shouldn't get another card until the debt is paid off or the creditors may cancel my low interest rates because they see that I'm seeking credit instead of "in a hardship" which got me the low rate in the first place. I apply anyway for a home improvement card from Wells Fargo which will pay for the air conditioning. I don't get approved for enough CL to pay for it, so the HVAC company calls and gets them to raise my limit to exactly the amount of the central AC install. Boom, immediately maxed out $4200 card to add to the pile! My score is 610 at this time. It's got 3 years of no interest, so I pay extra on the payments to make sure that I pay it off in time before the 3 years is up. This is the first time that I've ever paid more than minimums on a credit card. I cross my fingers that my other creditors won't raise my rates.
About a year after the WF card, I find this forum. I also join the personal finance forum on reddit. It's eye opening. Maybe I could use credit the right way this time? Maybe I could earn rewards instead of paying interest? I can't wait to get started, so even though I've still got 8 months to pay off the old debts, and 2 years left to pay off the AC, I apply for new credit cards. Discover It was my first app, and it came back with an approval for a $10,000 limit! Holy cow! This is double what any of my highest limits have ever been! My score is 698 at this time. I also app for a PNC cash card and an Amex BCP. Those give me a $1,500 and $1,400 limit respectively, and bring me back down to earth a little. 
I pay them off every month. I pay zero in interest.
A few months go by, and I start getting obsessed with my credit score and realize that WF card high utilization is killing my score, so I call and see if they'll raise the limit. They approve a raise to $10,000! I take my 60 day AMEX CLI on the BCP and get it to $4,200. I app a couple more rewards cards and wait 6 more months for another CLI on the AMEX card. Boom, $12,600 with the next 3x CLI! I'm on a roll here! That card went from $1,400 to $12,600 in just 8 months! I finish paying off the old debt. I put extra towards the WF HVAC card to try to pay it off sooner. I open a savings account for the first time in my life and put the cashback in there. My credit score raises to around 750. Maybe 3rd time with credit cards will be the charm for me..
tl;dr:
- Discover is one of the best for a high starting limit even when your credit report isn't so hot.
- AMEX is one of the best for getting a high limit via CLI due to their 60 day 3x CLI followed by every 6 months 3x CLI.
I opened my BOFA Cash Rewards card in 2013 with a 1K SL. 2 years later in June 2015, I called for a CLI (back then it was a HP) and got approved for 10K.
