We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$-destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$-positive set which has finite intersection with every $A\in \mathcal {I}\cap V$. Also, we discuss the associated variants

$$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\},\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} $$

Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb {P}_I$ can $+$-destroy a Borel ideal $\mathcal {J}$; (2) we discuss many classical examples of Borel ideals, their $+$-destructibility, and cardinal invariants; (3) we show that the Mathias–Prikry, $\mathbb {M}(\mathcal {I}^*)$-generic real $+$-destroys $\mathcal {I}$ iff $\mathbb {M}(\mathcal {I}^*)\ +$-destroys $\mathcal {I}$ iff $\mathcal {I}$ can be $+$-destroyed iff $\mathrm {cov}^*(\mathcal {I},+)>\omega $; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$-generic real $+$-destroys $\mathcal {I}$, and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$-destroys $\mathcal {I}$; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.