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As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi \colon E \to B$ has (if it exists) degree $2^t$ for some $t \geq 0$. Let us say that $\xi$ is of type $t$ if $w_{2^t}$ is the smallest non-trivial SW-class (and of type $\infty$ if all SW-classes vanish).

The tangent bundles of $\mathbb R\! P^2, \mathbb C\! P^2, \mathbb H\! P^2$ and the Cayley plane are of type $0, 1, 2$ and $3$, respectively.

Stong determined the cohomology of the connected covers $BO\langle k \rangle$ of $BO$ (published in Trans. Am. Math. Soc. 107, 526-544 (1963).) This calculation implies that exponentially many (in $k$) SW-classes vanish, but still all but finitely many $w_i$ survive along the map induced from $BO\langle k \rangle \to BO$. From this, one can easily deduce the existence of vector bundles of arbitralily big finite type.

Presumably, this can be upgraded to also get tangent bundles of manifolds of arbitralily big finite type. But what are concrete examples of such manifolds?

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    $\begingroup$ Renee Hoekzema has done some work related to this, see e.g. people.math.rochester.edu/faculty/doug/otherpapers/Hoekzema.pdf where the Rosenfeld plane $(\mathbb{O}\otimes\mathbb{H})\mathbb{P}^2$ is discussed as a candidate of a manifold of type $4$ (it has type at least $4$ as far as I understood). $\endgroup$
    – Mark Grant
    Commented Jul 18, 2022 at 17:22
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    $\begingroup$ The total Stiefel-Whitney class of $\mathbb{RP}^k$ is $(1+x)^{k+1}$, where $x$ is the generator of $H^1(\mathbb{RP}^k; \mathbb Z/2)$. So when $k = 2^{t+1} + 2^t - 1$, $T\mathbb{RP}^k$ has type $t$. Is this the sort of example you had in mind? $\endgroup$
    – Arun Debray
    Commented Jul 25, 2022 at 15:59

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