Say you really like the company, people, tech stack, whatever and are interested in working there but you have other offers for more that you are currently leaning towards, but only because they are higher.

If you have a preferred element of compensation, suggest they increase that specifically. Don't ask what the max they are able to go to is as you'll rarely get a true hard limit that way, and once that happens negotiation is over.

Be prepared to at least *look* like you're willing to walk, even if it is the job you want.

I recently took my 2nd lowest initial offer up to my highest one by playing it off against two others and getting all three to the max they were able to offer me.

Wow, that's sick! If you don't mind me asking, did the other recruiters know you had offers and how much they were? I don't have any other offers right now so I'm banking on "I'm interviewing with other companies" instead. Not sure if that will work.