Is there an outdated or insecure hash algorithm thats output can be easily reversed back into the input?
Depending on what you want to do, maybe you can take a cryptographic hash function and truncate it. Adjust the length of the truncation to how much effort you want the preimage calculation to be. This is a valid way to create a proof-of-work mechanism. For example, if you want to require a party to perform about $2^{m+n}$ operations where $2^m$ is the estimated cost of hashing a very short string, then:


Generate a random $s+n$-bit string and calculate its hash $h$.
Send the hash $h$ and the first $s$ bits of the string and request a preimage.
Read back a string and check that its hash is $h$.


(The salt length $s$ needs to be sufficiently long to prevent pre-computing possible inputs.)

With any cryptographic hash function in good standing, and even with most of the ones in bad standing such as MD5, the fastest way to find a preimage is brute force searching among all $n$-bit strings.

If proof-of-work is not what you're after, then maybe you can use a non-cryptographic hash, for example a CRC. In general, hash functions that aren't specifically designed to be cryptographic hashes have a low probability of accidental collision (but it's easy to craft a collision deliberately), and they're easy to invert.

Of course, “inverting” the hash will find a preimage, not necessarily the string you started with.