What is a skill check and what does the d20 represent?
If the d20 represents "how well I did it this time", then what your players are doing makes some sense, especially if players have an idea of how well they are doing. Restricting players to "1 roll per player" doesn't really help here.
This, however, leads to boredom; pointless rolls.
Roll for difficulty
An alternative is that the d20 represents "how hard the task actually is". When you roll d20+STR vs DC 15, and you get a 1, it means that while most doors in this dungeon are medium-difficult to open, this particular door is exceedingly hard. In this case, a reroll only occurs if the situation is "reset" in a fundamental way.
If someone offers help?
Aid just modifies that existing roll. So if your STR is +5, you need to find +9 points of modifiers to open that door. Aid another? That is worth +2. A +5 luck bonus from the Bard? Almost there. Potion of giant strength granting +3 more strength? Finally, the door opens!
Repeated rolls under this system, where the roll reveals how hard something really is, don't make sense. DCs in this system are just rules of thumb the DM is following; a "typical" door is DC 15, to find out how hard this particular door is you roll your d20.
What more, the information revealed by the door being hard to open can influence later checks. If the door was unopenable because it was swelled from water, trying to pick it after won't do much good; but a spell that dries wood could. If it was unopenable because it had metal-reinforced bars, picking the lock might help.
Roll for skill
If the original approach holds -- the d20 represents how well the player tries -- then you have to decide when to ask the player to roll.
A player should only roll when there are consequences to the roll.
If there is no time pressure and failure means you waste time, then there are no consequences to the roll.
If the task is impossible, but a bad failure will cause damage, then there are consequences to the roll.
In general, the state of the game after a roll shouldn't be one such that "I try again" would always make sense. If it is, consider failing forward.
Imagine an impossible task. You decide that on an DC20 check you fail, but you learn it is impossible; if you roll 19 or under, you take 3d6 damage and learn that it is impossible. This is a check with consequences and "I try again" doesn't make sense.
Now, imagine a dangerous task. On a DC20 check you succeed, on failure you take 2d6 damage and can try again. In some circumstances this is reasonable, with the failure damage generating tension.
But it might be better to fail forward. DC20 you succeed. DC 15 you succeed after taking 2d6 damage. Under DC15 you take 2d6 damage and can choose to either take another 2d6 and succeed, or break it.
This is "fail forward" -- you fail, pay a price, and you proceed forward in the plot anyhow. It even has a gradient of failure (or a gradient of success depending on how you define it).
Notice I offered a bargain above. They could choose to give up on the obstacle, or pay a price and pass it. This could be overly metagamey for you.
Example: "I loot the bodies"
Going back to the critters. You kill a bunch of Orcs. You then do a scavenge check to see what they have of value.
Under "roll determines difficulty", your roll actually determines what is worth scavenging. On a low roll, stuff was destroyed or lost prior to finding it. On a high roll, more stuff was there to be found. Your scavenging skill factors in, but your efforts are assumed to be maximized; you find everything you can find.
Under "fail forward" with a bargain, maybe you say "you find 28 silver: If you want to risk getting a disease I'll give you another scavenge check." Without the bargain, the poor scavenge check might just give you lice (a minor disease), but regardless you find the decent quality sword owned by the orcs.
Under "only roll when it matters", have a fixed "scavenge skill to loot" result. Unless the players are pressed for time (say are being chased), in which case you can roll to determine how fast they can get the loot ("It's taking you a while to search; do you really want to when the worgs are after you?").