A Baer *-ring is an *-algebra whose lattice of projections is complete. I know two well-handed kinds of these structures:
1- W*-algebras (abstract case of von Neumann algebras).
2- The inverse limit of W*-algebras (called locally W*-algebra).
Q. Does there exist any other type(s) of Baer *-rings?
More examples. Take any field $F$ with a positive definite involution (that is, $\sum a_ia_i* = 0$ implies all $a_i = 0$). Form the ring of $n \times n$ matrices, $M_n F$, equipped with $*$-transpose; it is a Baer* ring. Moreover, we can take the bounded subring of $F$, $F_b$, consisting of those elements of $F$, $a$, for which we can solve $aa^* +\sum x_i x_i^* = m$ for some integer $m$). Then $M_n F_b$ contains all the projections of $M_n F$, hence is itself a Baer* ring.
For example, let $F = K(x_{\alpha})$, equipped with the identity involution, where $K$ is any subfield of the reals, and $\{x_{\alpha}\}$ is an arbitrary family of indeterminates (I wanted to give a field which could not be embedded in the complexes; choose the cardinality of the set of indeterminates to exceed that of the complexes).
And still more. Let $R$ be a self-injective von Neumann regular ring with positive definite involution. Then $R$ and all its matrix rings (and matrix rings over the bounded subrings as defined above) are Baer*. This includes the regular ring of AW* algebras of finite type, and also weird examples constructed from the field of fractions of the Weyl algebra.
I am not sure whether your definition of a Baer *-ring is correct. In the book of Berberian on Baer *-rings a ring $A$ is called a Baer *-ring if
1) $A$ is a *-ring (not necessarily an algebra), i.e., it admits an involution $a\mapsto a^*$ satisfying $a^{**}=a$, $(a+b)^*=a^*+b^*$, and $(ab)^*=b^*a^*$.
2) For each non-empty subset $S\subseteq A$ the right annihilator $R(S)=\{a\in A: sa=0\text{ for all }s\in S\}$ equals $pA=\{pa:a\in A\}$ for some projection $p\in A$.
The C*-algebra $C([0,1])$ of continuous functions from the unit interval to the complex numbers has only two projections, hence its projections form a complete lattice, but is certainly not a Baer *-algebra.
The Baer$\phantom{.}^*$-rings that are C*-algebras are called AW$^*$-algebras. The most prominent examples are of course von Neumann algebras, but there are many AW$^*$-algebras that are not von Neumann algebras, see for instance the book of Saito and Wright on monotone complete C*-algebras, a subclass of AW$^*$-algebras.
There are also examples of Baer$\phantom{.}^*$-rings that are neither AW$^*$-algebras nor locally W$^*$-algebras. Let $F$ be a finite field. For cardinality reasons, it cannot be isomorphic to an AW$^*$-algebra nor a locally W$^*$-algebra. Since its multiplication is commutative, the identity is an involution. If $S\subseteq F$ is non-empty, then it is easy to see that $R(S)$ is an ideal. Since $F$ has only trivial ideals, we have either $R(S)=(0)$, whence $R(S)=0F$, or $R(S)=F$, whence $R(S)=1R$. So $F$ is a Baer *-ring.
If $\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})$ is a finite von Neumann algebra, then the algebra $\mathrm{Aff}(\mathcal{M})$ of closed, densely defined operators affiliated with $\mathcal{M}$ (most easily defined as commuting with all of the unitaries of $\mathcal{M}'$). It turns out that $\mathrm{Aff}(\mathcal{M})$ is a von Neumann regular Baer $*$-ring containing $\mathcal{M}$, and is actually equal to the maximal ring of right (or left) quotients of $\mathcal{M}$. See this paper of Berberian for details.
