That is a LOT of concrete. Can you attach some kind of safety railing or fencing to the vertical sides? Plants could then be added to soften the look at the base, but plants alone will not provide a safety feature.

I would be inclined to install a guardrail with low voltage lighting along with some planting for safety reasons.
I would also demolish the steeply angled section, install a guardrail, night lighting and plant.
Someone could really injure themselves. This appears to be a case where spending money now will save you money and anguish later.

Attached is a guardrail / railing detail that would work in your situation.

I am astounded this got passed by a building inspector. Agree with deviant and catkim this is definitely a safety issue. I'll let the experts suggest possible planting ideas. My only thought on that is a climber to soften the top of the guard fencing.

I think your photos are too close-up. Can you provide some better photos that show the limits of this thing and how it relates to things that are located in its vicinity?

Thank you for the suggestions. I've considered railing, but the husband is opposed (possibly because of aesthetics, possibly because of the cost). Demo-ing the downward slanting portion of the ramp will cost at least $500 if we had someone come out to do it. I need to do some research to see if any step of a demo like that could be a DIY project. I'm sure some of it would need to be contracted out, but if we could save somewhere, we sure would like to.
Here are the requested pictures.

This is the driveway from the road. This is a corner property with the driveway behind the home. You can see there's a long stretch that could be planted with similar trees or shrubs. The property line extends about 4 feet beyond the driveway to the right.
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Here's another view:
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Another with my beauties for height perspective. I circled the utilities box. The end of this specimen of a ramp literally shimmies right up against the property line. That utilities box is technically the neighbor's yard. It's a shame that the previous owner built it like this!
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And here is the edge. I could see about putting some edible shrubs here that the neighbor and I both agree on, since this would technically be his land. It's a possibility he might be interested, but we definitely want to be cautious and good neighbors.
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We also considered adding soil and terracing this area, but since we have about four horizontal feet to work with, I'm not sure how well that would work.

To react to your proposal about solving this problem by planting shrubs around it ... it doesn't seem like the way to go in my opinion. For one thing, the shrubs will create a misleading illusion that there is not a drop-off or that it is not as severe, adding potentially more danger to the unwary (or unaware.) Another thing, using shrubs to hide or camouflage sometimes (fairly frequently actually) ends up creating an even bigger monstrosity (visually speaking.) If it were mine, I'd first be looking for any possible uses and readiest ones seem like something for wheeled entertainment ... skates ... BMX, etc. But this would require designing and ADDING to the monstrosity in order to finish it off by making it into something useful. One would need to conclude that they really wanted/needed such a thing, which is sounds like you're not apt to conclude, before they spent any money on extending it. The third thing that occurs, since this thing is concrete, do you need any retaining walls anywhere on your property? This section of drive could be broken up into manageable, fairly uniform pieces and used as retaining wall material. It would require the rental of a jackhammer and some labor. Cheaper, but not free. You'd also need to cart away (or figure out how to use) the dirt pile that will be under this drive.

(BTW, the last pictures added were more useful for comprehending what it is you are dealing with.)

Thank you. We're new to this, so I really do appreciate the suggestions. Obviously, we'd like to find a simple fix, but this one doesn't look to be one of those. I have a hill on the side of the house that we've considering terracing, but I think a retaining wall that tall would be overkill, especially since that would mean I'd have another four foot obstacle to keep the children from jumping off.

Now you've got me scratching my head... I'm sure I'll follow up with some more questions after I stew around this for a bit. Thanks.

I just don't see any way to fix this without putting up a railing of some kind around that crazy thing. It does not need to be fancy, just enough visible and heft to keep the cars and kids from going off the edges. I think it could even be a diy gig and not as costly as it looks.

"...but I think a retaining wall that tall would be overkill, especially since that would mean I'd have another four foot obstacle to keep the children from jumping off."My goodness, you would not use the material just as it stands now. It would be broken up and reconfigured into a new wall that is the height your design specifies. Broken concrete walls, with their broken side exposed, look basically like stone once they age a little. The "stone" can be laid dry-stack or mortared, just like regular stone. But if obtaining concrete from an existing structure to re-use in another, one must plan out how it will be broken up... such as in 12" x 18" blocks. The picture should give you the idea that is has potential for re-use in other structures. Google-image broken or recycled concrete to see other examples of it.

Yardvaark, this is exactly why I ask questions on gardenweb. What a great, helpful, creative answer to a portion of my dilemma. I didn't think of doing something like that, but I'm so thankful that YOU did! Thank you for the suggestion. We're a little closer to some decisions now, and we might just be demo-ing a part of that wall, and this type of retaining wall would be perfect for another project. So thank you, thank you, thank you. Great.

Also, I found some railing that would work and that wouldn't be too expensive. So I think we'll be a go on that.

Fair warning. Come Fall, I will probably be asking some questions about shrubs and trees again because, well, as first time homeowners, I'm all about getting shrubs and trees in the ground, like yesterday.

Thanks again for all of the helpful answers!

If you demo the ramp, there would be a logical way to do it and you would need to do a little preliminary exploration to discover its construction. (Dig down to see how deep it is. ) You'd likely demo the drive portion first and then knock over the "wall" portions and jack hammer them as they lie on the ground (instead of doing it while they're in the upright position.) If I were doing this, I'd mark out a grid (like 12" x 18" pieces) with a chalk line and break the slabs into uniform size pieces rather than random rubble.

The only real solution isn't cheap or easy. Remove the entire extension on the driveway, ramp, concrete retaining walls, and all. Re-landscape the area after you're finished.

The way it is now, it's a monstrosity that's just waiting for a serious accident and/or a lawsuit.

Possible temporary fix to build a wooden privacy fence around it, right at the outside edges of the driveway extension and ramp, with no gaps to fall into, nor a lower wall or curb to trip over. A privacy fence would be tall enough to either prevent such accidents or limit the extent of most injuries and accidents. of course, you're talking about additional concrete for the fenceposts, now...