- Configuration languages (JSON, YAML, XML) are pure combinatorial logic.
- Regular expressions are... mostly not actually regular, but you get the idea.
- Some templating languages are deliberately less powerful than Turing machines, e.g. ST4 is context-free.
- Prepared SQL statements are a similar idea on a different axis.
The real question is whether a non-TC language could be useful for general purpose programming. Such a language might come with very strong guarantees (termination, time complexity, even correctness or a limited form of correctness), but they might be extra-cumbersome for 'normal' workloads.
The bytecode is theoretically turing-complete, but the verifier present in the kernel ensures that the code contains no loops and accesses no out-of-bounds memory.
With `clang`, you can compile C down to eBPF bytecode. It's a subset of C, with a lot of restrictions, but it is familiar and capable of some remarkable things past just simple packet parsing & filtering.
Brainfuck is a classical example of a turing complete language that doesn't have a notion of functions. What does "recursion" mean here? Nothing, I think. So you don't need recursion for Turing completeness.
Some people work on languages that support functions as first class objects, which don't natively support recursion. This is nice from a language design standpoint, because functions only see the scope of their arguments and can't look any further. The "Y Combinator" pattern is used to bootstrap recursion into these languages. You may have heard of it.
Turing-incomplete languages can also support unlimited recursion. For example, take a Turing complete language, and equip it with a state of the art theorem prover. Before entering a loop, or calling a function, the theorem prover is allowed to spend up to 1 minute proving that the loop or function call will terminate. If the theorem prover times out, the program halts with an error. Some functions, such as a naive implementation of factorial, can be easily proved to halt for all inputs. So this language allows recursion, but by construction, all programs terminate, so it's turing incomplete.
Obviously using a recursive descent parser is out but thats probably easy to work around right?
I could be wrong tho so don't base mission critical code on this without rigorous testing!
Key to being a good programmer: at least read a bit about the language before speculating about it. Not sure what you mean by "allows constants", but you're only given increment and decrement operators, and a promise that fresh tape is zeroed out.
Brainfuck supports iteration. Performing recursion in a higher level language will still look like iteration in Brainfuck.
Making a language that have the expressive power of finite state machines could be an example.