He is asking about the most effective good deeds in expiating sins

Firstly: 

Allah, may He be exalted, expiates sins by means of repentance, calamities,
good deeds that erase sin, and other means. 

Ibn al-Qayim (may Allah have mercy on him) said: The traces of sin may be
erased by means of sincere repentance, heartfelt affirmation of the Oneness
of Allah (Tawheed), good deeds that erase sin, calamities that expiate sin,
the intercession of those who intercede among those who affirm the Oneness
of Allah, and finally, if he is punished for whatever sins remain, he will
be brought out of Hell by virtue of his Tawheed (affirmation of the Oneness
of Allah).

End quote from Hidaayat al-Hayaari, p. 130 

Good deeds that erase sin include wudoo’, the five daily prayers, and Hajj
and ‘Umrah. The greatest of good deeds is Tawheed (affirmation of the
Oneness of Allah), and the gravest of evil deeds is kufr (disbelief) and
shirk (ascribing partners to Allah). 

Saying Laa ilaaha ill-Allaah (there is no god but Allah) is the
greatest of good deeds, and the most sublime of the branches of faith, as
the Prophet (blessings and peace of Allah be upon him) said: “Faith has
seventy-odd – or sixty-odd – branches, the best of which is saying Laa
ilaaha ill-Allaah, and the least of which is removing something harmful from
the road, and modesty is a branch of faith.”

Narrated by al-Bukhaari (9) and Muslim (35), from the hadeeth of Abu
Hurayrah. 

Muslim narrated in his Saheeh (2687) that Abu Dharr said: The
Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah,
may He glorified and exalted, said: Whoever does a good deed will have a ten
fold reward and more, and whoever does a bad deed its recompense is one like
it, or I will forgive him.” 

It was narrated from Abu Sa‘eed al-Khudri (may Allah be pleased with him)
that the Messenger of Allah (blessings and peace of Allah be upon him) said:
“Moosa (peace be upon him) said: ‘O Lord, teach me something by which I may
remember You and call upon You.’ He said: ‘Say: Laa ilaaha ill-Allaah
(there is no god but Allah).’ He said: ‘O Lord, all Your slaves say that.’
He said: ‘Say: Laa ilaaha ill-Allaah.’  He said: ‘There is no god but
You, my Lord, but I wanted something that is only for me.’ He said: ‘O Moosa,
if the seven heavens and all their inhabitants others except Me, and the
seven earths, were (placed) in one pan (of the Balance) and Laa ilaaha
ill-Allah was placed in the other pan, Laa ilaaha ill-Allah would
outweigh them.”

Narrated by Ibn Hibbaan (6218) and al-Haakim (1936). Al-Haakim classed it as
saheeh and adh-Dhahabi agreed with him. Al-Haafiz said in al-Fath
(11/208): It was narrated by an-Nasaa’i with a saheeh isnaad 

This points to the virtue of these great words. 

Ibn Rajab (may Allah have mercy on him) said:

If a person’s Tawheed (affirmation of the Oneness of Allah) and his
sincerity towards Allah are perfect, and he fulfills all its conditions in
his heart, on his lips (verbally) and in his physical actions, or in his
heart and on his lips at the time of death, then forgiveness for all his
previous sins becomes certain, and it will protect him from entering Hell
altogether. When a person truly believes in Tawheed in his heart, and he
excludes from that everything other than Allah in terms of love, veneration,
glorification, awe, fear, hope and trust, in that case his sins will all be
erased, even if they are like the foam of the sea, and they may be turned
into good deeds. For this Tawheed is the greatest elixir; if an atom of it
were to be placed on mountains of sins, it would turn them into good deeds.

End quote from Jaami‘ al-‘Uloom wa’l-Hukam, 2/417

 Undoubtedly those who utter these great words will vary in status,
depending on how deeply its various meanings take root in their hearts. 

Ibn al-Qayyim (may Allah have mercy on him) said: 

The greater a person’s Tawheed (affirmation of the Oneness of Allah), the
more complete will Allah’s forgiveness of him be. Whoever meets Him not
associating anything with Him at all, He will forgive him all his sins, no
matter what they were, and he will not be punished for them. 

We do not say that none of the people of Tawheed will enter the Fire; rather
many of them will enter it because of their sins and will receive punishment
commensurate with their wrongdoing, then they will be brought out of it.
There is no contradiction between the two matters for the one who properly
understands what we have mentioned above. 

Here we will explain further the greatness of this status, because it is
essential that it be understood. It should be noted that the rays of Laa
ilaaha ill-Allah scatter the fog and clouds of sin in accordance with
the strength or weakness of those rays. So this phrase has light, and those
who believe in it vary with regard to the strength or weakness of that
light, which no one knows except Allah, may He be exalted. … The greater the
light of these words, the more they will dispel specious arguments and whims
and desires, until the individual reaches a state where he does not
encounter any specious argument, whim, desire or sin, but these rays will
dispel it. This is the state of one who is actually sincere in his Tawheed
and does not associate anything with Allah.

End quote from Madaarij as-Saalikeen, 1/338 

Secondly: 

The words of Allah (interpretation of the meaning), “And whosoever fears
Allah and keeps his duty to Him, He will make a way for him to get out (from
every difficulty). And He will provide him from (sources) he never could
imagine” [at-Talaaq 65:2] offer a great promise to the pious who fear
Allah. 

The one who fears Allah is the one who fulfills obligatory duties and avoids
things that are forbidden. That includes the one who sins, then repents and
turns to Allah, and the one who goes back to sin, then repents from it
again, because piety or fearing Allah, may He be exalted, means doing what
is enjoined and refraining from what is forbidden. If a person repents
sincerely, his bad deeds will be expiated and turned into good deeds, as
Allah, may He be exalted, says (interpretation of the meaning):

“And those who invoke not any other ilaah (god) along with Allah, nor kill
such person as Allah has forbidden, except for just cause, nor commit
illegal sexual intercourse
and whoever does this shall receive the punishment.

69. The torment will be doubled to him on the Day of Resurrection, and he
will abide therein in disgrace;

70. Except those who repent and believe (in Islamic Monotheism), and do
righteous deeds; for those, Allah will change their sins into good deeds,
and Allah is Oft‑Forgiving, Most Merciful”

[al-Furqaan 25:68-70]. 

It is not stipulated that the pious one who fears Allah should be infallible
and free of bad deeds, otherwise most people would be deprived of this
honour and virtue. For every son of Adam sins, and the best of those who sin
are those who repent. The one who sins then repents is one of the pious for
whom there is the hope of the promise mentioned in the verse quoted above.
The grace of Allah is vast, and what matters is that each individual should
focus on striving. 

And Allah knows best.