He suggests me___(ask)the teacher for help.
答案asking.
Why?
为什么不用虚拟语气？

答案肯定是对的，只是suggest sb doing出现的频率不是很高，一般很少用 更多的时候使用suggest sb's doing，相当于动名词的复合结构，即等价于 suggest doing sth.和suggest (that) sb. (should) do sth.

suggest 是个及物动词，后面可以跟名词、代词、动名词或从句作它的宾语。各种试题对 suggest 的考查通常涉及到它的非谓语动词形式、宾语从句及宾语从句的虚拟语气等方面。

一、 suggest 作“间接表明；建议；使想起”解时，后面可以跟名词、代词或动名词做宾语

① suggest sth.表明，建议
That girl's sun-tanned face suggests excellent health.

那个姑娘被太阳晒黑了的脸表明她身体非常健康。

Mr. Smith suggested a way out of the difficulty to her.

史密斯先生向她提出了一个克服困难的办法。

② suggest doing sth

Tom suggested selling the house but Ann was against it.

汤姆建议把房子卖了，但安不同意。

二、 suggest后面的宾语从句要不要用虚拟语气，由它的意思决定： 作“建议”解，宾语从句中的谓语动词用虚拟语气，即谓语动词用“（ should ）动词原形”。

　　① Almost all his friends suggested that he visit Mr. Johnson at once.

　　② ----The experiment has failed！

----I suggest you try again.

句式一、 It's suggested that…… （有人建议……）例如：

　　① It's suggested that we （ should ） put on a short play at the party.

有人建议我们在晚会上演一个短剧。

　　② It has been suggested that Robert should go to Chicago and try to find a job there. 有人建议罗伯特到芝加哥去，设法在那里找个工作。

　　句式二、 What （ All that ） sb. suggests is that…… （某人建议的是……）

　　① What I suggest is that we visit a class right away.

我的建议是我们马上去听课。

　　② What she suggested was that the clinic should make better use of those herbs. 她建议医疗站更好地利用那些草药。

suggest 作“表明、说明、暗示、指出”解，后面的that 宾语从句的谓语动词用陈述语气。

① He often came to the theater， but his attitude suggested he was not really interested in the play.

　　② Jane's pale face suggested that she was ill， and her parents suggested that she have a medical examination.

　　第一个 suggested 作“表明、说明”解，所以其后的宾语从句的谓语动词用陈述语气；第二个 suggested 的意思是“建议”，所以其后的宾语从句的谓语动词用虚拟语气，其中的 should 被省略了。

至于上文中为什么不用虚拟语气，上文已经写得很清楚

suggest解释建议时，后面的宾语从句要用虚拟语气，suggest (that) sb. (should) do sth.这里的sb需要用人称代词主格

联系到上文，suggests为谓语动词，me为宾格，asking为宾语补足语，是一句简单句，没有出现宾语从句这一语法结构，因此不用虚拟语气,

综上，答案asking显然是正确的