
anwachsen
to increase
To increase in amount, number, extent, or intensity; to become larger.
Der Schuldenberg ist auf eine beachtliche Summe angewachsen.
The mountain of debt has increased to a considerable sum.
to grow / increase
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to increase
To increase in amount, number, extent, or intensity; to become larger.
Der Schuldenberg ist auf eine beachtliche Summe angewachsen.
The mountain of debt has increased to a considerable sum.

to take root
Refers to plants that, after being transplanted, take root in a new location and begin to grow.
Trotz des trockenen Sommers sind fast alle jungen Bäume gut angewachsen.
Despite the dry summer, almost all the young trees took root well.

to grow onto
To become firmly attached to a surface or another object through growth.
Der Efeu ist so fest an der Hauswand angewachsen, dass man ihn kaum entfernen kann.
The ivy has grown so firmly onto the house wall that it can hardly be removed.
'Anwachsen' (to grow/increase) is a separable verb. The prefix 'an-' appears at the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Druck auf die Regierung wächst täglich an.
The pressure on the government is increasing daily.
The perfect tense of 'anwachsen' (to grow/increase) is always formed with the auxiliary verb 'sein' (to be), because it describes a change of state or growth. So the correct form is 'ist angewachsen' (has grown/has increased).
Die Bevölkerung der Stadt ist im letzten Jahr stark angewachsen.
The population of the city increased sharply last year.
When the result of the growth is a specific number or amount, the preposition 'auf' is used followed by the accusative.
Das Vermögen des Unternehmens ist auf über eine Milliarde Euro angewachsen.
The company's assets have increased to over one billion euros.
This meaning is used in gardening and botany. It describes whether a plant successfully takes root in a new place. It is often combined with adverbs like 'gut' (well) or 'schlecht' (poorly).
Die frisch gepflanzten Rosen sind gut angewachsen.
The freshly planted roses have taken root well.
In this meaning, the verb describes a firm, physical connection with a surface. It is almost always used with the preposition 'an' and the dative to name this surface.
Die Muscheln sind fest am Schiffsrumpf angewachsen.
The mussels have grown firmly onto the ship's hull.
to rise / climb
„Steigen“ and „anwachsen“ both describe an increase. „Steigen“ is especially suitable for measurable numbers, values, or levels. „Anwachsen“ presents something more as a growing amount or scope, although the two uses partly overlap.

Der Mitgliederbestand wächst jeden Monat an.
The membership base grows every month.
Die Zahl der Mitglieder steigt jeden Monat.
The number of members increases every month.