A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition.

In fact, in any group with binary operation, say +, the identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2).

There are plenty of groups with elements a satisfying a + a = 0. ℤ₂ as mentioned above has its unique non-zero element of order 2. The Klein group V₄ has three non-identity elements, each of order 2. Dihedral groups D₂ₙ (the symmetry groups of regular n-gons) contain reflections, all of which have order 2. Symmetric groups Sₙ (n ≥ 2) contain transpositions, each of which has order 2.

For example, in the dihedral group D₈, if we let a be a reflection of the square, then a + a = 0 and a + a + a = a. But this is conventionally written in multiplicative notation as a² = the identity element, so a³ = a.

Similarly, in the symmetric group S₃ under the binary operation of composition, if a denotes the transposition (12), then (12)(12) is the identity element and (12)(12)(12) = (12). In other words, applying a transposition three times is the same as applying it once.

In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically. It is conventional to use + in some subjects (coding theory, additive groups of integers modulo n, etc.) and · in others (permutation groups, dihedral groups, etc.). Often + is used for the binary operation in abelian groups and · in non-abelian ones. I'm sure none of this is particularly insightful to someone who has studied group theory, but still I wanted to share a few concrete examples here.

> A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition.

That’s a good example of an algebra where 1 + 1 + 1 = 1, but the article is specifically about systems where in addition to that condition, this second condition is also true: 1 + 1 != 0 (not equal!). ℤ₂ is not an example of that.

[My post below is wrong]

> In fact, in any group with binary operation +, identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2).

The "if" is correct. The "only if" is not. (I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation", as I don't recall cases where "+" and "*" are used for specific types of binary operations).

> The Klein group V₄ has three non-identity elements, each of order 2.

Unrelated, but this calls out for a link to the classic song Finite Simple Group (of Order Two) by the Klein Four: https://www.youtube.com/watch?v=BipvGD-LCjU