Here's an excerpt from Lasenby, Lasenby and Doran, 1996, A Unified Mathematical Language for Physics and Engineering in the 21st Century:
The next crucial stage of the story occurs in 1878 with the work of
the English mathematician, William Kingdon Clifford (Clifford 1878).
Clifford was one of the few mathematicians who had read and understood
Grassmann's work, and in an attempt to unite the algebras of Hamilton
and Grassmann into a single structure, he introduced his own
geometric algebra. In this algebra we have a single geometric product
formed by uniting the inner and outer products—this is
associative like Grassmann's product but also invertible, like
products in Hamilton's algebra. In Clifford's geometric algebra an
equation of the type $\mathbf{ab}=C$ has the solution
$\mathbf{b}=\mathbf{a}^{-1}C$, where $\mathbf{a}^{-1}$ exists and is
known as the inverse of a. Neither the inner or outer product
possess this invertibility on their own. Much of the power of
geometric algebra lies in this property of invertibility.
Clifford's algebra combined all the advantages of quaternions with those of
vector geometry, [...]
Every element of a geometric algebra can be identified with a tensor, but not every tensor can be identified with an element of a geometric algebra.
It's helpful to consider the vector derivative of a linear operator, of a map from vectors from vectors. Call such a map $\underline A$. The vector derivative is then
$$\partial_a \underline A(a) = \partial_a \cdot \underline A(a) + \partial_a \wedge \underline A(a) = T + B$$
where $T$ is a scalar, the trace, and $B$ is a bivector. The linear map $\underline A$ can then be written as
$$\underline A(a) = \frac{T}{n} a + \frac{1}{2} a \cdot B + \underline S(a)$$
where $\underline S$ is some traceless, symmetric map. While the scalar can be turned into a multiple of the identity, in $T \underline I/n$, and the bivector can be directly turned into an antisymmetric map in $a \cdot B$, the map $\underline S$ is very much part of $\underline A$, yet not representable in general through a single algebraic element of the geometric algebra. This is just one example of such an object.
Best Answer
You know that blades are directed measures. $a\wedge b$ for instance is a directed area, $a\wedge b\wedge c$ is a directed volume, so I suppose what you're asking is: what is for instance $1 + a + a\wedge b + a\wedge b\wedge c$?
It's a difficult question and I'm not sure anyone has a definite answer, but I suspect it's the same kind of question people often ask about a paravector $1 + a$ the first time they see one : "how can you add a scalar to a vector? That doesn't make any sense!"
Every time I'm confronted to this question I remember what a math teacher of mine told our class once in order to explain why scalar multiplication is noted $\lambda v$ and not $v\lambda$ : "people say I have two carrots, not I have carrots two". Ever since I've heard this I think of vector spaces as grocery stores. A vector is a bag in which you put stuff that can be very different. Apples and oranges are totally different things so you keep separate numbers to count them, but you can still put them in the same bag.
I hope that helps.