Is there a method that gets beneficial diminishing returns when adding more dice, yet stays random?
You're actually asking two questions:

How can I design a function that computes a result on a dice roll that gives asymptotically decreasing benefit to adding more dice?

How can I have a system where adding more dice doesn't lead to a predictable result?


The first question is simple.  A trivial answer is to take the highest die, or highest X dice.  Alternatively, set some threshold that you need to roll above, such as roll at least one six, or roll a total of at least 15.  Under any of these schemes, as you roll more dice, you are more likely to have dice with better results, but there is a fundamental limit on how high you can go.  In the case of highest X dice, your limit is the highest value on those dice, and in the case of targets, the limit is a 100% chance of success.
By way of a simple example, let's say you use D6's, and you're taking the highest die.  Your chance of getting a 6 (the best possible result) looks like this:

But while this satisfies your goal of the first question, it is directly contradictory to the goal of your second question.  As you add more dice, the result gets more predictable.  This is a general principle in probability; the more independent random events you aggregate, the lower the collective variance.
Given this, I believe that the answer to your second question is that &quot;it's impossible without a re-framing of the question&quot;.  Adding more dice will decrease the variance of the result, and therefore increase the predictability.
The way to re-frame the question is to not have all of the dice be the same.  Here's a simple example, to illustrate the concept, though you likely will want to do something more complex.
You have two kinds of dice, white and red.  White dice contribute to success, red represent risk.  Each item, skill, etc adds some amount of each dice.  Most will add small amounts of white dice (1 or 2), but some better items may add larger amounts of white dice (3 - 5) but some red dice as well (1 - 2).  When you roll the dice, you take the highest result of the white dice... unless any of the red dice came up as a &quot;1&quot;, in which case you treat it as a 0.  Through a system like this, you have to make trade-offs between getting increasingly marginal benefit from more white dice vs the risk of catastrophic failure from an additional red die.  Maybe even have everyone roll 1 red die at baseline just for fun.
That's the type of idea you're probably looking for.  Obviously you have to make it your own and fit it to your game.  You can get more interesting if you take the top N dice instead of just the top 1.  You could add a third kind of die.  The list goes on.