I am in charge, monthly of organizing potluck luncheons, for my water aerobics groups.

A favorite is:
BEST DARN HAM SANDWICHES YOU'LL EVER HAVE!!!
2 -12 packages of sweet Hawaiian rolls (the small dinner roll looking ones)
1 1/2 lbs of Virginia Ham (NOT honey ham, it is a little too sweet)
12 slices Swiss Cheese
1 stick of real butter
2 teaspoons Worcestershire sauce
1 teaspoon Garlic Powder or Minced Garlic
1 teaspoon Onion Powder, or Minced Onion
1 teaspoon poppy seeds or Sesame Seeds

Directions:
You will need two 9 x 13 pans. Place the bottoms of 12 rolls in each pan. Place ham (about 2 shaved slices or so) on the rolls. Cut the cheese slices into 4 parts and place 2 small pieces on each sandwich. Put the dinner roll tops on. In a sauce pan, mix butter, Worcestershire sauce, onion powder, garlic powder and poppy seeds. Wait until all butter is melted and then brush the melted mixture over the ham sandwiches. Cover with foil and let sit in fridge for 1 hour or over night. (If you want to bake them right away, you can also.) Preheat oven to 375 and bake for 15 minutes or until cheese is melted. Serve. They are great hot and even at room temperature Enjoy! PLEASE Remove the FOIL for Baking ~ it makes them toasty

I am thinking but man you are really limiting the possibilities.
It can't go in the fridge, that's a biggie, no crock pots or appliances, needs to be a main dish, and can hold up for several hours.
I was considering a big bowl of bagged salad and some diced up chicken in it, like rotisserie chicken shredded or grilled chicken. You can buy the bags of fully cooked fajita chicken strips. If you could put the bag of salad and the bags of cooked chicken in the fridge or a small portable cold bag with blue ice in it. That could work. Have a large bowl to dump it all in, mix it up and have unopened bottles of dressing.
One way or another you are going to have to lug something there.

If you can't refrigerate, I would keep it vegetarian/non-dairy just to be safe, though cheese is usually okay for a few hours.

Some ideas:
Oil and vinegar based pasta salad with lots of veggies

Greek salad with cheese tortellini, cucumbers, olives, red onions, artichoke hearts, feta cheese

Pinwheel sandwiches made with hummus and veggies

Skewers with salami, artichoke hearts, olives, fresh mozzarella drizzled with balsamic or pesto

Black bean, corn and avocado salad (slice the avocado up right before lunch and add)

Caprese salad with slices of tomato, fresh mozzarella and basil drizzled with balsamic

Glenda's idea is probably your best bet right now. If you were not limited to just a main dish I would have tons of ideas, but I am kind of drawing a blank. Any kind of a big sub sandwich would most likely work.

Do you have a cooler? How about quiche?

I always used to take Chinese chicken salad. I would prepare all of the ingredients at home, but package and transport them separately in a cooler with blue ice. Right before lunch, I would toss all of the ingredients together with the dressing in a huge plastic bowl I found at Costco and serve.

There are those bags, I have 2, that are thermal bags, large enough to fit a casserole.

I use them to transport cold OR hot dishes and they keep for a number of hours and no problem.

Have an upcoming potluck this coming Monday, and I'm taking shrimp quiche.

I make an awesome tomato pie that I've taken to potluck and it can be served cold, or at room temp.

You could make something in a medium sized bowl and place it into a larger bowl that has a layer of ice in it, to keep it cold.

I think some kind of salad with meat would be great. There are never enough salads or veggies for my taste. :)

Yes I am worried about the no cold storage.
I think no question that you should take some kind of cooler or thermal bag with dry ice. We have several of the big thermal bags from Costco but I have a smaller one from big lots I use with blue ice to carry a few bottles of water on our garden club bus trips.
Something along those lines should be easy enough to transport

Do you have any method to heat up a dish at work?

Walmart sells giant sub sandwiches for a little over $5. You could pick up a few of them and cut into single serve sizes. Or make your own but I think those should be kept cool.

Dollar Tree sells the thermal bags, I have several and they work!

I'm with Glenda. Take a cooler or cooler bag, put those frozen ice packs in it and take anything you want. Curried chicken salad, Glenda's ham sandwiches, shrimp and pasta salad. I'm not sure what the big deal is about a crockpot. You could make up a taco salad, put the dressing on just before serving and bring a crockpot with the meat mixture in it, turn it on low when you get to the office and just put it in the salad before serving. Sounds easy to me. I did it for about 60 ladies one year when the golf course neglected to tell me their restaurant was going to be closed for renovations when our member-guest day was scheduled. We had a Mexican theme so it was the best idea I could come up with one week before. Turned out to be the best luncheon ever!

Alice, what kinds of things did you have in mind that requires no chilling, no cooking on site?

With those limitations, I'd really consider cold sandwiches of some sort. Surely you can tote a small cooler for a variety of sliced meats and cheeses. Buy small containers of condiments so that they can be transported unopened, and a nice variety of breads and rolls.

I know, it's a real dilemma. We have a couple fridges at work, but they are always so full, that there is no room for a large casserole dish. Our only way to heat food is microwaves. I am not willing to haul my personal, very nice crock-pot to and from work with my commute. Also, space for plugging it in and keeping it safe from children is limited, and I am not hauling a hot crock-pot five minutes across the campus when it is time to eat. I cannot have it going in my classroom either.

I do have cooler packs and thermal bags! :)

I LOVE the idea of a salad, but my co-workers are super carnivores. Salad goes mostly uneaten. So, sandwiches or quiche sound like the front runners. Costco sells quiches in 2 packs, so, I think I'll bake a couple quiches, keep them cold, and then serve room temperature.

Thank you all VERY MUCH for the ideas!!! :)

If you keep them refrigerated or cooled before serving. I make these a lot and they get gobbled up like candy.

Ham and cheese sandwiches:

hamburger buns-the over-sized ones
cream cheese-some with chive flavored cream cheese
ham
cheese (use various presliced cheeses

quick, easy, portable

I make them and cut them in half so someone not wanting a full one doesn't have to eat them.

One time, someone brought in a huge Subway sandwich for potluck. He bought it the night before and kept it out on his kitchen counter all night, (too long to put in the fridge) brought it in at 7:30 in the morning. Potluck wasn't until 12:00 or 1:00. Some people ate it, I didn't. Had mayo, meat and cheeses on it. Gross.