How can I get the depth of a multidimensional std::vector at compile time?

A classic templating problem. Here's a simple solution like how the C++ standard library does. The basic idea is to have a recursive template that will count one by one each dimension, with a base case of 0 for any type that is not a vector.

#include <vector>
#include <type_traits>

template<typename T>
struct dimensions : std::integral_constant<std::size_t, 0> {};

template<typename T>
struct dimensions<std::vector<T>> : std::integral_constant<std::size_t, 1 + dimensions<T>::value> {};

template<typename T>
inline constexpr std::size_t dimensions_v = dimensions<T>::value; // (C++17)

So then you could use it like so:

dimensions<vector<vector<vector<int>>>>::value; // 3
// OR
dimensions_v<vector<vector<vector<int>>>>; // also 3 (C++17)

Edit:

Ok, I've finished the general implementation for any container type. Note that I defined a container type as anything that has a well-formed iterator type as per the expression begin(t) where std::begin is imported for ADL lookup and t is an lvalue of type T.

Here's my code along with comments to explain why stuff works and the test cases I used. Note, this requires C++17 to compile.

#include <iostream>
#include <vector>
#include <array>
#include <type_traits>

using std::begin; // import std::begin for handling C-style array with the same ADL idiom as the other types

// decide whether T is a container type - i define this as anything that has a well formed begin iterator type.
// we return true/false to determing if T is a container type.
// we use the type conversion ability of nullptr to std::nullptr_t or void* (prefers std::nullptr_t overload if it exists).
// use SFINAE to conditionally enable the std::nullptr_t overload.
// these types might not have a default constructor, so return a pointer to it.
// base case returns void* which we decay to void to represent not a container.
template<typename T>
void *_iter_elem(void*) { return nullptr; }
template<typename T>
typename std::iterator_traits<decltype(begin(*(T*)nullptr))>::value_type *_iter_elem(std::nullptr_t) { return nullptr; }

// this is just a convenience wrapper to make the above user friendly
template<typename T>
struct container_stuff
{
    typedef std::remove_pointer_t<decltype(_iter_elem<T>(nullptr))> elem_t;    // the element type if T is a container, otherwise void
    static inline constexpr bool is_container = !std::is_same_v<elem_t, void>; // true iff T is a container
};

// and our old dimension counting logic (now uses std:nullptr_t SFINAE logic)
template<typename T>
constexpr std::size_t _dimensions(void*) { return 0; }

template<typename T, std::enable_if_t<container_stuff<T>::is_container, int> = 0>
constexpr std::size_t _dimensions(std::nullptr_t) { return 1 + _dimensions<typename container_stuff<T>::elem_t>(nullptr); }

// and our nice little alias
template<typename T>
inline constexpr std::size_t dimensions_v = _dimensions<T>(nullptr);

int main()
{
    std::cout << container_stuff<int>::is_container << '\n';                 // false
    std::cout << container_stuff<int[6]>::is_container<< '\n';               // true
    std::cout << container_stuff<std::vector<int>>::is_container << '\n';    // true
    std::cout << container_stuff<std::array<int, 3>>::is_container << '\n';  // true
    std::cout << dimensions_v<std::vector<std::array<std::vector<int>, 2>>>; // 3
}

Assuming that a container is any type that has value_type and iterator member types (standard library containers satisfy this requirement) or a C-style array, we can easily generalize Cruz Jean's solution:

template<class T, typename = void>
struct rank : std::integral_constant<std::size_t, 0> {};

// C-style arrays
template<class T>
struct rank<T[], void> 
    : std::integral_constant<std::size_t, 1 + rank<T>::value> {};

template<class T, std::size_t n>
struct rank<T[n], void> 
    : std::integral_constant<std::size_t, 1 + rank<T>::value> {};

// Standard containers
template<class T>
struct rank<T, std::void_t<typename T::iterator, typename T::value_type>> 
    : std::integral_constant<std::size_t, 1 + rank<typename T::value_type>::value> {};

int main() {
    using T1 = std::list<std::set<std::array<std::vector<int>, 4>>>;
    using T2 = std::list<std::set<std::vector<int>[4]>>;

    std::cout << rank<T1>();  // Output : 4
    std::cout << rank<T2>();  // Output : 4
}

Container types can be further restricted if needed.

You can define the following class template vector_depth<> which matches any type:

template<typename T>
struct vector_depth {
   static constexpr size_t value = 0;
};

This primary template corresponds to the base case that ends the recursion. Then, define its corresponding specialization for std::vector<T>:

template<typename T>
struct vector_depth<std::vector<T>> {
   static constexpr size_t value = 1 + vector_depth<T>::value;
};

This specialization matches an std::vector<T> and corresponds to the recursive case.

Finally, define the function template, GetDepth(), that resorts to the class template above:

template<typename T>
constexpr auto GetDepth(T&&) {
   return vector_depth<std::remove_cv_t<std::remove_reference_t<T>>>::value;
}

Example:

auto main() -> int {
   int a{}; // zero depth
   std::vector<int> b;
   std::vector<std::vector<int>> c;
   std::vector<std::vector<std::vector<int>>> d;

   // constexpr - dimension determinted at compile time
   constexpr auto depth_a = GetDepth(a);
   constexpr auto depth_b = GetDepth(b);
   constexpr auto depth_c = GetDepth(c);
   constexpr auto depth_d = GetDepth(d);

   std::cout << depth_a << ' ' << depth_b << ' ' << depth_c << ' ' << depth_d;
}

The output of this program is:

0 1 2 3