The problem is knowing what is and isn't really healthy. Reading labels doesn't do you any good, companies can get away with anything. Reese's cereal is "part of a healthy breakfast" after all. And, most of what is "common knowledge" on healthy eating is wrong advice from the debunked low-fat theories of the 60s and 70s.
The best advice remains - eat real food and avoid processed foods as much as possible. Millions of species do pretty well sticking with what nature gives us and many of our real diet based health problems started with the agricultural revolution. Bread is not a natural food.
Fruits and many veggies come in handy snack sized portions, are tasty, not generally messy, and you can mindlessly munch as much as you want.
Avoid low-fat cheeses, they are generally terrible for you. Full fat cheeses are better, just don't eat too much. But a few cubes of cheese set out with some awesome fruit or veggies and hummus makes good snacking. Crackers are as bad, and often worse, than chips.
Nuts contain much needed healthy fats. Snacking too mindlessly on these can bust a calorie count pretty easily, but if you keep a limited supply and eat them with fruit, you're snacking well.
Popcorn is good. Personally, I dislike air-popped, just too bland. But, you can oil-pop with some good butter and/or healthy oils like olive or peanut (avoid oils from non-oily plants like soybean, canola, "vegetable" blends). Butter is not evil, it's a whole food, healthy saturated fat, but it does pack a load of calories. Still, a tablespoon is 100 calories, popped into a whole pot of popcorn shared among 4-6 people... that's pretty negligible but oh so tasty.
Sushi is not bad or expensive to make for the game group. I do this all the time. Sushi rice in bulk is cheap. Ingredients for a cali roll are cheap, too. I add a bit of something fresh I can get the day of the game, some tuna, yellowtail, scallops, whatever, to make some nigiri and/or throw into a roll or two. Once you get the rice made, making the rolls is easy and doesn't take much time.
Another snack I make often is chicken wings. I roast them, not deep fried, and a good basic buffalo sauce has no carbs, not too much sodium, and is great snacking. Doesn't meet the not-messy rule though, by a long shot.