i wasn't able to get a runnable forth to less than a couple of pages written in itself https://github.com/kragen/stoneknifeforth but schönfinkel's ski-combinators are maybe the simplest practical basis

    s f g x → f x (g x)
    k a b   → a
    i = s k k (one of many possible definitions)

maybe wolfram knows of a simpler alternative

my favorite is abadi and cardelli's object-calculus on which i based bicicleta. it has two reduction rules rather than the λ-calculus's one. using a hybrid of bicicleta's syntax and abadi and cardelli's

    {f = ς(v)b, g = ς(v)c, ...}{f = ς(v)d} → {f = ς(v)d, g = ς(v)c, ...}
    {f = ς(v)b, g = ς(v)c, ...}.f          → b[{f = ς(v)b, g = ς(v)c, ...}/v]

the first of these derives an inherited object with a new definition for method f. the second one invokes method f on an object, which is evaluated by replacing its self-variable v with the object itself throughout its body, using the standard β-reduction semantics with α-renaming that we're familiar with from the λ-calculus (b[x/y] means b but with x replacing y)

despite the simplicity of the semantics the ς-calculus is enormously more usable for actually writing down functions than the λ-calculus. here's factorial(10) in the notation above (untested)

    {fac = ς(env){
        n = ς(_)3,
        return = ς(o)
            (o.n < 2).if_true{
                then = ς(_)1
                else = ς(_)o.n * env.fac{n = ς(_)o.n - 1}.return
            }.return
        }
    }.fac{n = ς(10)}.return

bicicleta has a lot of syntactic sugar which reduces this to (tested)

    {env: fac = {fac:
        arg1 = 3
        '()' = (fac.arg1 < 2).if_true(
            then = 1
            else = fac.arg1 * env.fac(fac.arg1 - 1)
         )
    }}.fac(10)

in particular to be able to define infix operators inside the language, bicicleta rewrites

    x * y

as

    x.'*'{arg1 = y}.'()'

and analogously for -, <, etc.

they published a bunch of papers and a book on this but they were more interested in static typing than anything else. a paper on one imperative variation of the thing is http://lucacardelli.name/Papers/PrimObjImpSIPL.A4.pdf

you can implement a turing machine in a few lines of c, making it internally simpler than the other alternatives, but as you point out it's unusable except as a compilation target or to prove theorems about