Remaking std::to_string in C++ with UDLs (User Defined Literals)
I'm ignoring all the testing stuff, and just focusing on the two conversion functions.

std::string operator"" _str(unsigned long long int num)
{
    std::deque&lt;char&gt; digits;


There's really no need for the deque. You can push_back() all the digits directly into a std::string, and then use std::reverse().

To make it even better, you could use the max value of an unsigned long long and calculate the maximum number of digits you'll need, then use reserve(). But that probably won't net you any real gains in reality, since you'll likely always be under the small-string optimization size where std::string won't actually allocate any memory... another reason to ditch the deque and stick with strings.

for(char i = 0; num != 0;num = num / 10)


Have you tested your conversion functions with zero? Because if num is zero, this loop won't fire at all, and digits will be empty. So "0"_str will result in "".

This is a pretty weird and confusing loop in general. i looks like it's going to be some kind of loop counter, but it's not. All-in-all, this just looks like a case of trying to be too clever, and outsmarting yourself (due to the zero issue).

This could be expressed much more simply and clearly as something like:

do
{
    digits.push_front('0' + (num % 10));
    num /= 10;
} while (num != 0);


and it won't have the zero bug. (do loops are less common than while, but the reason to use one here - and this should be documented in a comment - is to guarantee that even if num is zero, the loop will fire at least once.)

{snipped section about digit character codes}

std::string converternum;


I already mentioned that the deque is unnecessary, but if you're really going to use it, you should take advantage of the fact that you know digits.size(), and use reserve() for this string to prevent unnecessary allocations.

char * operator"" _cstr(unsigned long long int num)


This function is mostly identical to the first one, and thus has the same problems. Again, I would ditch the deque and just work with a string. The digits will be in reverse order, but in this case you can fix that with std::reverse_copy() when you copy them into your output buffer.

char * string = (char *)converternum.c_str();
return string;//converternum.c_str();


Here is where you get into real trouble.

First, never, ever use C-style casts. In this case, you've cast away the constness of c_str()... that's bad. Prior to C++17, if you want a non-const pointer to std::string's data, you have to jump through some hoops - you have to do something like &amp;convertnum[0] or &amp;convertnum.front() - and I agree that's a pain. But casting away constness is a red flag that won't pass most code reviews, because it is extremely dangerous.

Second, you are taking a pointer to convertnum's internal data, and returning that... even though convertnum is being destroyed. In other words, you are returning a pointer to freed memory, which is UB.

How do you solve these problems? Well, the don't answer is: Don't use C-style strings. In other words, don't have a _cstr UDL. It's only going to give you grief (it already has!).

But if you really, really must... one option is to do what @hoffmale suggests, and instead of char*, return std::unique_ptr&lt;char[]&gt;. That might look something like this:

char* operator"" _cstr(unsigned long long int num)
{
    auto digits = std::string{};
    do
    {
        digits += '0' + (num % 10);
        num /= 10;
    } while (num != 0);

    auto p = std::make_unique&lt;char[]&gt;(digits.size() + 1);

    std::reverse_copy(digits.cbegin(), digits.cend(), p.get());
    p[digits.size()] = '\0'; // have to manually add the null terminator

    return p;
}


The problem with that is you're now longer returning a C-string, which makes "_cstr" kind of a lie.

So another option is to have some kind of storage with a lifetime longer than the function that will retain the memory for you. Something like:

char* operator"" _cstr(unsigned long long int num)
{
    // Note: you *can't* use a vector&lt;string&gt;, because string's
    // memory may be invalidated on moves, like if the vector
    // resizes.
    static auto storage = std::vector&lt;std::unique_ptr&lt;char[]&gt;&gt;{};

    auto digits = std::string{};
    do
    {
        digits += '0' + (num % 10);
        num /= 10;
    } while (num != 0);

    auto p = std::make_unique&lt;char[]&gt;(digits.size() + 1);

    std::reverse_copy(digits.cbegin(), digits.cend(), p.get());
    p[digits.size()] = '\0';

    // Get the char*, and then move the unique_ptr into storage
    // to let it be managed there.
    auto result = p.get();
    storage.push_back(std::move(p));

    return result;
}


This will "work", and it's basically what some C libraries do (when they don't simply leak). But with no way to release the memory, it will just grow and grow. You could add a mechanism to release the memory, but you would need a way to be sure it's safe for every string. This solution is possible, but unwieldy.

The last solution - the worst case (other than outright UB) - is to define some sort of contract that client code must follow. In plain English, document the function by saying "you must delete[] the memory when you're done with it". That requires programmers to behave, and is very easy to break, so it is not recommended. But... if you really want a C-string function, it's an option. A bad one, but an option nonetheless.

Summary

You have one critical error in your code:


In ""_cstr(unsigned long long), you return a pointer to the internal memory of a std::string that gets destroyed at the end of the function, meaning you're returning a pointer to invalid memory.


You have a couple of logical errors in your code:


Both functions will return an empty string when the number is zero, instead of "0".


You have a few inefficiencies:


You use deques because you want to add the numbers in reverse order, and reason that makes sense because neither strings nor vectors have push_front(). But that's all unnecessary. You can simply push_back() directly into your result string, and then use std::reverse(). (Or push_back() into a string buffer, and std::reverse_copy() into your final buffer.)


And here are some tips to keep in mind:

It's really good that you're doing testing, but if you're going to do that, you might as well go all in and use something like Google Test. It's header-only, a single include, and it produces very pretty output. A full battery of tests can give you great confidence in your code. Using Google Test might look something like (fair warning, I don't actually use Google Test that often):

TEST(StrTest, HandlesZero) {
    EXPECT_EQ(0_str, "0");
}


To test that the types are right, rather than trying to demangle the names (which, by the way, it's been a long time since I've used abi::__cxa_demangle(), but I'm pretty sure you're not using it correctly), it's so much easier to just use type traits:

TEST(StrTest, ReturnType) {
    EXPECT_TRUE((std::is_same_v&lt;decltype(0_str), std::string&gt;));
}


Another dirty trick you can use to examine types is deliberately generating errors:

template &lt;typename T&gt;
struct TD; // note, never define TD

TD&lt;decltype(0_str)&gt; t; // will trigger an error


The error generated will print the exact type of whatever is in the angle brackets, on every compiler. This is a hackish way to check types, but it can help when you're experimenting.