CRC32 at compile time proof of concept. This can be expanded to virtually any other hash, and probably adapted to be cleaner in design (using the hash function as a template argument).

constexpr crc32 proof of concept.cpp

#include <cstdint>

namespace {
    /// IEEE CRC32 table.
    static constexpr std::uint32_t crc32_table[] = {
        0x00000000, 0x77073096, 0xEE0E612C, 0x990951BA,
        0x076DC419, 0x706AF48F, 0xE963A535, 0x9E6495A3,
        0x0EDB8832, 0x79DCB8A4, 0xE0D5E91E, 0x97D2D988,
        0x09B64C2B, 0x7EB17CBD, 0xE7B82D07, 0x90BF1D91,
        0x1DB71064, 0x6AB020F2, 0xF3B97148, 0x84BE41DE,
        0x1ADAD47D, 0x6DDDE4EB, 0xF4D4B551, 0x83D385C7,
        0x136C9856, 0x646BA8C0, 0xFD62F97A, 0x8A65C9EC,
        0x14015C4F, 0x63066CD9, 0xFA0F3D63, 0x8D080DF5,
        0x3B6E20C8, 0x4C69105E, 0xD56041E4, 0xA2677172,
        0x3C03E4D1, 0x4B04D447, 0xD20D85FD, 0xA50AB56B,
        0x35B5A8FA, 0x42B2986C, 0xDBBBC9D6, 0xACBCF940,
        0x32D86CE3, 0x45DF5C75, 0xDCD60DCF, 0xABD13D59,
        0x26D930AC, 0x51DE003A, 0xC8D75180, 0xBFD06116,
        0x21B4F4B5, 0x56B3C423, 0xCFBA9599, 0xB8BDA50F,
        0x2802B89E, 0x5F058808, 0xC60CD9B2, 0xB10BE924,
        0x2F6F7C87, 0x58684C11, 0xC1611DAB, 0xB6662D3D,
        0x76DC4190, 0x01DB7106, 0x98D220BC, 0xEFD5102A,
        0x71B18589, 0x06B6B51F, 0x9FBFE4A5, 0xE8B8D433,
        0x7807C9A2, 0x0F00F934, 0x9609A88E, 0xE10E9818,
        0x7F6A0DBB, 0x086D3D2D, 0x91646C97, 0xE6635C01,
        0x6B6B51F4, 0x1C6C6162, 0x856530D8, 0xF262004E,
        0x6C0695ED, 0x1B01A57B, 0x8208F4C1, 0xF50FC457,
        0x65B0D9C6, 0x12B7E950, 0x8BBEB8EA, 0xFCB9887C,
        0x62DD1DDF, 0x15DA2D49, 0x8CD37CF3, 0xFBD44C65,
        0x4DB26158, 0x3AB551CE, 0xA3BC0074, 0xD4BB30E2,
        0x4ADFA541, 0x3DD895D7, 0xA4D1C46D, 0xD3D6F4FB,
        0x4369E96A, 0x346ED9FC, 0xAD678846, 0xDA60B8D0,
        0x44042D73, 0x33031DE5, 0xAA0A4C5F, 0xDD0D7CC9,
        0x5005713C, 0x270241AA, 0xBE0B1010, 0xC90C2086,
        0x5768B525, 0x206F85B3, 0xB966D409, 0xCE61E49F,
        0x5EDEF90E, 0x29D9C998, 0xB0D09822, 0xC7D7A8B4,
        0x59B33D17, 0x2EB40D81, 0xB7BD5C3B, 0xC0BA6CAD,
        0xEDB88320, 0x9ABFB3B6, 0x03B6E20C, 0x74B1D29A,
        0xEAD54739, 0x9DD277AF, 0x04DB2615, 0x73DC1683,
        0xE3630B12, 0x94643B84, 0x0D6D6A3E, 0x7A6A5AA8,
        0xE40ECF0B, 0x9309FF9D, 0x0A00AE27, 0x7D079EB1,
        0xF00F9344, 0x8708A3D2, 0x1E01F268, 0x6906C2FE,
        0xF762575D, 0x806567CB, 0x196C3671, 0x6E6B06E7,
        0xFED41B76, 0x89D32BE0, 0x10DA7A5A, 0x67DD4ACC,
        0xF9B9DF6F, 0x8EBEEFF9, 0x17B7BE43, 0x60B08ED5,
        0xD6D6A3E8, 0xA1D1937E, 0x38D8C2C4, 0x4FDFF252,
        0xD1BB67F1, 0xA6BC5767, 0x3FB506DD, 0x48B2364B,
        0xD80D2BDA, 0xAF0A1B4C, 0x36034AF6, 0x41047A60,
        0xDF60EFC3, 0xA867DF55, 0x316E8EEF, 0x4669BE79,
        0xCB61B38C, 0xBC66831A, 0x256FD2A0, 0x5268E236,
        0xCC0C7795, 0xBB0B4703, 0x220216B9, 0x5505262F,
        0xC5BA3BBE, 0xB2BD0B28, 0x2BB45A92, 0x5CB36A04,
        0xC2D7FFA7, 0xB5D0CF31, 0x2CD99E8B, 0x5BDEAE1D,
        0x9B64C2B0, 0xEC63F226, 0x756AA39C, 0x026D930A,
        0x9C0906A9, 0xEB0E363F, 0x72076785, 0x05005713,
        0x95BF4A82, 0xE2B87A14, 0x7BB12BAE, 0x0CB61B38,
        0x92D28E9B, 0xE5D5BE0D, 0x7CDCEFB7, 0x0BDBDF21,
        0x86D3D2D4, 0xF1D4E242, 0x68DDB3F8, 0x1FDA836E,
        0x81BE16CD, 0xF6B9265B, 0x6FB077E1, 0x18B74777,
        0x88085AE6, 0xFF0F6A70, 0x66063BCA, 0x11010B5C,
        0x8F659EFF, 0xF862AE69, 0x616BFFD3, 0x166CCF45,
        0xA00AE278, 0xD70DD2EE, 0x4E048354, 0x3903B3C2,
        0xA7672661, 0xD06016F7, 0x4969474D, 0x3E6E77DB,
        0xAED16A4A, 0xD9D65ADC, 0x40DF0B66, 0x37D83BF0,
        0xA9BCAE53, 0xDEBB9EC5, 0x47B2CF7F, 0x30B5FFE9,
        0xBDBDF21C, 0xCABAC28A, 0x53B39330, 0x24B4A3A6,
        0xBAD03605, 0xCDD70693, 0x54DE5729, 0x23D967BF,
        0xB3667A2E, 0xC4614AB8, 0x5D681B02, 0x2A6F2B94,
        0xB40BBE37, 0xC30C8EA1, 0x5A05DF1B, 0x2D02EF8D
    };
}

/// Bytes refers to a byte array passed into crc32.
/// BytesLength is the length of bytes.
/// BytesOffset is the offset being calculated.
/// Initial is the initial value.
///
/// Bytes is interpreted as std::uint8_t[BytesLength].
/// For char arrays `sizeof(my_char_array)` works.
///
/// Bytes, BytesLength, and Initial form the Tag.
///
/// This is recursive.
///
/// Remember that BytesLength is not just sizeof(somechars).
/// It is siezof(somechars)-1 (unless you stripped the NULL).
template<const char Bytes[], std::uint32_t BytesLength, std::uint32_t Initial = 0,
    const std::uint32_t* Table = crc32_table>
struct crc32 {
    /// Gets a pointer to the static bytes.
    static constexpr const char* get_bytes() {
        return Bytes;
    }

    /// Size of the Bytes stream.
    static constexpr std::uint32_t size() {
        return BytesLength;
    }

    template<std::uint32_t value, std::uint32_t offset>
    static constexpr std::uint32_t step() {
        // Inline test is to remove issues with recursion.
        constexpr std::uint32_t tmp =
            Table[(value ^ (std::uint8_t)Bytes[offset >= BytesLength ? 0 : offset]) & 0xFF] ^ (value >> 8);
        if (offset + 1 == BytesLength) {
            // End of CRC32 pass.
            return tmp ^ ~0U;
        }
        // This amazing hack resets the state to prevent
        // infinite recursion.
        return step<
            offset + 1 >= BytesLength ? Initial ^ ~0U : tmp,
            offset + 1 >= BytesLength ? 0 : offset + 1>();
    }

    // All we need is to step.
    static constexpr std::uint32_t get_value() {
        return step<Initial ^ ~0U, 0>();
    }
};

// "Test String" = 0x92779fa4
static constexpr char crc32_test_string[] = "Test String";

int main(int argc, char **argv) {
    // Remember to trim the null value off the end unless
    // your other crc32 system uses it too.
    static_assert(crc32<crc32_test_string, sizeof(crc32_test_string)-1, 0>::get_value() == 0x92779fa4, "crc32 failed");
    return 0;
}