A program that enumerates all theorems in ZFC and stops when it proves a contradiction (e.g. true = false). Encoding a program that is equivalent to that directly as a Turing Machine in merely 748 states: https://www.scottaaronson.com/papers/bb.pdf (meaning that we cannot prove an upper bound on the 748th Busy-Beaver number, but there are probably even smaller undecidable TMs).

But my favourite example (shown here in Java) demonstrates the difficulty of analysing simple, realistic programs without necessarily being undecidable:

    long foo(long x) {
        if (x <= 2 || (x & 1) != 0)
            return 0;
        for (var i = x; i > 0; i--)
            if (bar(i) && bar(x - i))
                return i;
        throw new Error();
    }
    
    boolean bar(long x) {
        for (var i = x - 1; i >= 2; i--)
            for (var s = x; s >= 0; s -= i)
                if (s == 0)
                    return false;
        return true;
    }

Even in this case where even the input space is finite (and so everything here is definitely decidable), we simply don't yet know whether there is some x for which foo(x) throws, let alone if we made the input unbounded by using BigInteger instead of long.