What does it mean to say that there doesn't exist an algorithm from a TM point of view?
First, note that it is easy to write algorithms that not always halt:

algo(n) {
  while ( n % 2 != 0 ) {
    n += 2
  }
  return n
}


It seems that you are asking for a semi-decidable problem of practical relevance. Consider verification of hard- and software. Many interesting properties require logics of higher order for their formulation, and usually satisfiability and/or validity of such formulae is undecidable. 

But these logics have proof system, sets of axioms and rules that make up every proof for valid formulae¹. Along these systems, all proofs can be (recursively) enumerated: thus, we can semi-decide many important properties. Using clever heuristics, lots of computational power and human intervention, e.g. by providing key lemmata, many problems can be solved in practice (i.e. for some useful instances) that are theoretically undecidable.




Beware incomplete proof systems.