For set-forcing, the answer is no, see http://arxiv.org/abs/1106.1951 by Hamkins, Kirmayer and Perlmutter.

The "Kunen Inconsistency" is the theorem that there is no nontrivial elementary embedding $j: V \to V$. The above article shows (among several other things) that if $V[G]$ is any set-forcing extension of $V$ then there is no nontrivial elementary embedding $j: V[G] \to V$. So even if you replaced ultrapowers by extenders, for example, the answer remains no.

For set-forcing, the answer is no, see the following article

Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter, Generalizations of the Kunen inconsistency, Ann. Pure Appl. Logic 163 (2012), no. 12, 1872--1890. (see also arxiv.org/abs/1106.1951 and Hamkins's blog post)

The "Kunen Inconsistency" is the theorem that there is no nontrivial elementary embedding $j: V \to V$. The above article shows (among several other things) that if $V[G]$ is any set-forcing extension of $V$ then there is no nontrivial elementary embedding $j: V[G] \to V$. So even if you replaced ultrapowers by extenders, for example, the answer remains no.

For set-forcing, the answer is no, see http://arxiv.org/abs/1106.1951 by Hamkins, Kirmayer and Perlmutter.

The "Kunen Inconsistency" is the theorem that there is no nontrivial elementary embedding $j: V \to V$. The above article shows that if $V[G]$ is any set-forcing extension of $V$ then there is no nontrivial elementary embedding $j: V[G] \to V$. So even if you replaced ultrapowers by extenders, for example, the answer remains no.

For set-forcing, the answer is no, see http://arxiv.org/abs/1106.1951 by Hamkins, Kirmayer and Perlmutter.

The "Kunen Inconsistency" is the theorem that there is no nontrivial elementary embedding $j: V \to V$. The above article shows (among several other things) that if $V[G]$ is any set-forcing extension of $V$ then there is no nontrivial elementary embedding $j: V[G] \to V$. So even if you replaced ultrapowers by extenders, for example, the answer remains no.