  In this paper, we study some new examples of ideals on $\omega$ with maximal
Tukey type (that is, maximal among partial orders of size continuum). This
discussion segues into an examination of a refinement of the Tukey order --
known as the "weak Rudin-Keisler order" -- and its structure when restricted to
these ideals of maximal Tukey type. Mirroring a result of Fremlin on the Tukey
order, we also show that there is an analytic P-ideal above all other analytic
P-ideals in the weak Rudin-Keisler.
  Acknowledgment: this preprint has not undergone peer review or any
post-submission improvements or corrections. The Version of Record of this
article is published in the Archive for Mathematical Logic, and is available
online at https://doi.org/10.1007/s00153-023-00897-z.
