I think it looks nice the way it is, but you could prune it up if you want to. Just do it gradually, one or two branches each year spread over a few years.

Also you don't want the lawn going right up to the tree anyway, it is important to leave a grass-free circle round the tree. Grass right up to the tree means less good growth (grass roots are greedy and make nutrients harder for the tree to get), and inevitable damaging mower crashes onto the tree's trunk.

Resin

I would wait till late fall to early winter to prune your tree. It looks like you have a double header on your tree,if so remove one, you should only have one central leader on any tree. Yes I would limb up your tree you want to be able to walk under your tree without hitting your head some day

That is NOT a bur oak.
Can't tell for sure from the photo, as I can't see leaf outlines clearly, but the habit and leaf morphology says sawtooth oak, Q.acutissima, to me.

Resin, I tried to kill the grass around the tree so I wouldn't have to fight with the limbs every time I mowed. Last year I let the bermuda get out of control. Is there a season that would be better for pruning an oak?
You are right lucky_p. It was labeled "burr oak" and I never even thought about checking the leaves after it grew enough. Thats what National Arbor Day Foundation does for a $10 donation. I've learned too many things about them to do that again.
Thanks tmore for drawing my attention to the double leader.

Can you get a close-up of the foliage, please?

Resin

Sawtooth oak has a frilly, ornate acorn cap kind of resembling that of bur oak, but I don't have any difficulty differentiating the caps, acorns, leaves or trees.
If the goobers at NADF can't make the differentiation - or they don't care that they're purposely sending out misidentified seedlings, they oughta get out of the business.

In any case, to answer one of the original questions - yes, you can limb it up. I would remove no more than a third - a quarter might be better - of the total leaf canopy at any one time - you don't have to, and probably shouldn't, do it all at any one time in any one year. So you might have to resign yourself to struggling with the riding mower for another few years. Widening the soon-to-be mulch circle might make it easier to mow - any more grass can be killed with Round-up or smothered with flattened cardboard or 4-6 sheets of black-ink newspaper under the mulch. Don't apply the mulch any deeper than 4", and put no mulch in the inch or so next to the trunk.

Removing branches now might invite bugs/borers to move in, so I might wait until the fall/winter to decide what to cut off. Follow the usual good pruning practices - sharp tools; clean cuts, flush with the branch collar; and leave no stubs.

Right now you have a bush, not a tree. Basically, every thing you see now is where it will always be. I would remove everything till you get a straight trunk. I would wait till the tree looses its leaves so you can get a good look at the shape. As the tree grows continue to remove lower limbs until you get a hight where you can stand under the tree. Do NOT put any type paint where you trim the limbs. The tree will heal itself. I agree with the others, this is not a burr oak.

{{gwi:394510}}
Here is the foliage.

Thanks! Yep, that's Sawtooth Oak Quercus acutissima. Native to eastern Asia.

Resin

Yeah, NADF can't seem to get it right -- one of the few survivors of a shipment from them I got is supposedly a Burr oak, but it looks like a typical E. White oak to me.

Thanks everyone for your identification of this tree and the pruning and mulching advice.
After examining the tree, I found that it doesn't have a double leader. The photo shows a branch lower on the trunk that comes up beside the main leader. Thats one less cut I'll have to make

TDSmith

td
A coworker gave me a bundle of 10-12 miniscule little sticks she'd gotten from NADF, along with their 'color-coded' ID list. Most look believable as what they claim them to be, but the ones that were supposed to be redbuds aren't and the ones that were supposed to be flowering dogwood did not have opposite leafing/branching, so I have no idea what they were - not dogwoods - but they didn't survive, anyway.

Non-native trees were being handed out for Arbor Day? That's irresponsible!

[ "Non-native trees were being handed out for Arbor Day? That's irresponsible!" ]

Yeah, I'm kinda disappointed too that they would choose to send out those.

Uh, yeah.
In the bundle of seedlings my coworker gave me were a couple of goldenraintrees(Koelreuteria) - talk about an invasive self-seeding alien - those went right into the trash bin - or at least what NADF said were goldenraintree seedlings went into the trash - they were just small bareroot seedlings with a gob of paint as their identifying mark.