For anyone reading who hasn't read the intro post in Feline Health,
here it is. Welcome to the Lantus/Lev/Basaglar forum! If you don't mind sharing, do you have a first name we can use, or do you just prefer we call you by your log in name?
I agree with Sienne, I think your kitty may have a secondary condition that is causing insulin resistance. For kitties who have either acromegaly - caused by a benign tumour on the pituitary that causes excess growth hormone to be produced, or IAA (insulin auto antibodies), sort of like an allergic reaction to injected insulin, these kitties can need higher doses of insulin than most others. My girl had both conditions. The good news is that for cats for one or other of these conditions, there is eventually an insulin dose that takes the cat into normal blood sugar numbers. We have a number of cats around there with those conditions that are fairly well regulated. Depending on which literature you read either 20 or 25% of all diabetic cats has acromegaly, so it's not uncommon. Though vets used to be taught it was rare, recent research proves otherwise.
You might want to start the conversation with your vet about getting blood tests done for these conditions. In North America, blood for these tests is sent to Michigan State University, the only place that does these tests here. Depending on the results of these tests, there may be some treatment options to help Kitty Boy.
As far as dosing goes, keep increasing every 3 days like you have, unless you see blue or you get to 10 units. If either of those happens, post here for more guidance. Or post more often if you'd like.